assignment 9 matrices

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Precalculus

Course: precalculus   >   unit 7, intro to matrices.

Matrix dimensions

B = [ − 8 − 4 23 12 18 10 ] ‍  

Check your understanding

• Your answer should be
• an integer, like 6 ‍  
• a simplified proper fraction, like 3 / 5 ‍  
• a simplified improper fraction, like 7 / 4 ‍  
• a mixed number, like 1   3 / 4 ‍  
• an exact decimal, like 0.75 ‍  
• a multiple of pi, like 12   pi ‍   or 2 / 3   pi ‍  

Matrix elements

• (Choice A)   [ 1 2 6 4 5 − 2 ] ‍   A [ 1 2 6 4 5 − 2 ] ‍  
• (Choice B)   [ − 9 6 7 − 3 − 3 5 ] ‍   B [ − 9 6 7 − 3 − 3 5 ] ‍  
• (Choice C)   [ 2 6 8 7 − 3 1 ] ‍   C [ 2 6 8 7 − 3 1 ] ‍  
• (Choice D)   [ 2 10 8 6 − 3 1 ] ‍   D [ 2 10 8 6 − 3 1 ] ‍  

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7.5 Matrices and Matrix Operations

Learning objectives.

In this section, you will:

• Find the sum and difference of two matrices.
• Find scalar multiples of a matrix.
• Find the product of two matrices.

Two club soccer teams, the Wildcats and the Mud Cats, are hoping to obtain new equipment for an upcoming season. Table 1 shows the needs of both teams.

A goal costs $300; a ball costs $10; and a jersey costs $30. How can we find the total cost for the equipment needed for each team? In this section, we discover a method in which the data in the soccer equipment table can be displayed and used for calculating other information. Then, we will be able to calculate the cost of the equipment.

Finding the Sum and Difference of Two Matrices

To solve a problem like the one described for the soccer teams, we can use a matrix , which is a rectangular array of numbers. A row in a matrix is a set of numbers that are aligned horizontally. A column in a matrix is a set of numbers that are aligned vertically. Each number is an entry , sometimes called an element, of the matrix. Matrices (plural) are enclosed in [ ] or ( ), and are usually named with capital letters. For example, three matrices named A , B , A , B , and C C are shown below.

Describing Matrices

A matrix is often referred to by its size or dimensions: m × n m × n indicating m m rows and n n columns. Matrix entries are defined first by row and then by column. For example, to locate the entry in matrix A A identified as a i j , a i j , we look for the entry in row i , i , column j . j . In matrix A ,   A ,   shown below, the entry in row 2, column 3 is a 23 . a 23 .

A square matrix is a matrix with dimensions n × n , n × n , meaning that it has the same number of rows as columns. The 3 × 3 3 × 3 matrix above is an example of a square matrix.

A row matrix is a matrix consisting of one row with dimensions 1 × n . 1 × n .

A column matrix is a matrix consisting of one column with dimensions m × 1. m × 1.

A matrix may be used to represent a system of equations. In these cases, the numbers represent the coefficients of the variables in the system. Matrices often make solving systems of equations easier because they are not encumbered with variables. We will investigate this idea further in the next section, but first we will look at basic matrix operations .

A matrix is a rectangular array of numbers that is usually named by a capital letter: A , B , C , A , B , C , and so on. Each entry in a matrix is referred to as a i j , a i j , such that i i represents the row and j j represents the column. Matrices are often referred to by their dimensions: m × n m × n indicating m m rows and n n columns.

Finding the Dimensions of the Given Matrix and Locating Entries

Given matrix A : A :

• ⓐ What are the dimensions of matrix A ? A ?
• ⓑ What are the entries at a 31 a 31 and a 22 ? a 22 ? A = [ 2 1 0 2 4 7 3 1 − 2 ] A = [ 2 1 0 2 4 7 3 1 − 2 ]
• ⓐ The dimensions are 3 × 3 3 × 3 because there are three rows and three columns.
• ⓑ Entry a 31 a 31 is the number at row 3, column 1, which is 3. The entry a 22 a 22 is the number at row 2, column 2, which is 4. Remember, the row comes first, then the column.

Adding and Subtracting Matrices

We use matrices to list data or to represent systems. Because the entries are numbers, we can perform operations on matrices. We add or subtract matrices by adding or subtracting corresponding entries.

In order to do this, the entries must correspond. Therefore, addition and subtraction of matrices is only possible when the matrices have the same dimensions . We can add or subtract a 3 × 3 3 × 3 matrix and another 3 × 3 3 × 3 matrix, but we cannot add or subtract a 2 × 3 2 × 3 matrix and a 3 × 3 3 × 3 matrix because some entries in one matrix will not have a corresponding entry in the other matrix.

Given matrices A A and B B of like dimensions, addition and subtraction of A A and B B will produce matrix C C or matrix D D of the same dimension.

Matrix addition is commutative.

It is also associative.

Finding the Sum of Matrices

Find the sum of A A and B , B , given

Add corresponding entries.

Adding Matrix A and Matrix B

Find the sum of A A and B . B .

Add corresponding entries. Add the entry in row 1, column 1, a 11 , a 11 , of matrix A A to the entry in row 1, column 1, b 11 , b 11 , of B . B . Continue the pattern until all entries have been added.

Finding the Difference of Two Matrices

Find the difference of A A and B . B .

We subtract the corresponding entries of each matrix.

Finding the Sum and Difference of Two 3 x 3 Matrices

Given A A and B : B :

• ⓐ Find the sum.
• ⓑ Find the difference.
• ⓐ Add the corresponding entries. A + B = [ 2 − 10 − 2 14 12 10 4 − 2 2 ] + [ 6 10 − 2 0 − 12 − 4 − 5 2 − 2 ] = [ 2 + 6 − 10 + 10 − 2 − 2 14 + 0 12 − 12 10 − 4 4 − 5 − 2 + 2 2 − 2 ] = [ 8 0 − 4 14 0 6 − 1 0 0 ] A + B = [ 2 − 10 − 2 14 12 10 4 − 2 2 ] + [ 6 10 − 2 0 − 12 − 4 − 5 2 − 2 ] = [ 2 + 6 − 10 + 10 − 2 − 2 14 + 0 12 − 12 10 − 4 4 − 5 − 2 + 2 2 − 2 ] = [ 8 0 − 4 14 0 6 − 1 0 0 ]
• ⓑ Subtract the corresponding entries. A − B = [ 2 −10 −2 14 12 10 4 −2 2 ] − [ 6 10 −2 0 −12 −4 −5 2 −2 ] = [ 2 − 6 −10 − 10 −2 + 2 14 − 0 12 + 12 10 + 4 4 + 5 −2 − 2 2 + 2 ] = [ −4 −20 0 14 24 14 9 −4 4 ] A − B = [ 2 −10 −2 14 12 10 4 −2 2 ] − [ 6 10 −2 0 −12 −4 −5 2 −2 ] = [ 2 − 6 −10 − 10 −2 + 2 14 − 0 12 + 12 10 + 4 4 + 5 −2 − 2 2 + 2 ] = [ −4 −20 0 14 24 14 9 −4 4 ]

Add matrix A A and matrix B . B .

Finding Scalar Multiples of a Matrix

Besides adding and subtracting whole matrices, there are many situations in which we need to multiply a matrix by a constant called a scalar. Recall that a scalar is a real number quantity that has magnitude, but not direction. For example, time, temperature, and distance are scalar quantities. The process of scalar multiplication involves multiplying each entry in a matrix by a scalar. A scalar multiple is any entry of a matrix that results from scalar multiplication.

Consider a real-world scenario in which a university needs to add to its inventory of computers, computer tables, and chairs in two of the campus labs due to increased enrollment. They estimate that 15% more equipment is needed in both labs. The school’s current inventory is displayed in Table 2 .

Converting the data to a matrix, we have

To calculate how much computer equipment will be needed, we multiply all entries in matrix C C by 0.15.

We must round up to the next integer, so the amount of new equipment needed is

Adding the two matrices as shown below, we see the new inventory amounts.

Thus, Lab A will have 18 computers, 19 computer tables, and 19 chairs; Lab B will have 32 computers, 40 computer tables, and 40 chairs.

Scalar Multiplication

Scalar multiplication involves finding the product of a constant by each entry in the matrix. Given

the scalar multiple c A c A is

Scalar multiplication is distributive. For the matrices A , B , A , B , and C C with scalars a a and b , b ,

Multiplying the Matrix by a Scalar

Multiply matrix A A by the scalar 3.

Multiply each entry in A A by the scalar 3.

Given matrix B , B , find −2 B −2 B where

Finding the Sum of Scalar Multiples

Find the sum 3 A + 2 B . 3 A + 2 B .

First, find 3 A , 3 A , then 2 B . 2 B .

Now, add 3 A + 2 B . 3 A + 2 B .

Finding the Product of Two Matrices

In addition to multiplying a matrix by a scalar, we can multiply two matrices. Finding the product of two matrices is only possible when the inner dimensions are the same, meaning that the number of columns of the first matrix is equal to the number of rows of the second matrix. If A A is an m × r m × r matrix and B B is an r × n r × n matrix, then the product matrix A B A B is an m × n m × n matrix. For example, the product A B A B is possible because the number of columns in A A is the same as the number of rows in B . B . If the inner dimensions do not match, the product is not defined.

We multiply entries of A A with entries of B B according to a specific pattern as outlined below. The process of matrix multiplication becomes clearer when working a problem with real numbers.

To obtain the entries in row i i of A B , A B , we multiply the entries in row i i of A A by column j j in B B and add. For example, given matrices A A and B , B , where the dimensions of A A are 2 × 3 2 × 3 and the dimensions of B B are 3 × 3 , 3 × 3 , the product of A B A B will be a 2 × 3 2 × 3 matrix.

Multiply and add as follows to obtain the first entry of the product matrix A B . A B .

• To obtain the entry in row 1, column 1 of A B , A B , multiply the first row in A A by the first column in B , B , and add. [ a 11 a 12 a 13 ] [ b 11 b 21 b 31 ] = a 11 ⋅ b 11 + a 12 ⋅ b 21 + a 13 ⋅ b 31 [ a 11 a 12 a 13 ] [ b 11 b 21 b 31 ] = a 11 ⋅ b 11 + a 12 ⋅ b 21 + a 13 ⋅ b 31
• To obtain the entry in row 1, column 2 of A B , A B , multiply the first row of A A by the second column in B , B , and add. [ a 11 a 12 a 13 ] [ b 12 b 22 b 32 ] = a 11 ⋅ b 12 + a 12 ⋅ b 22 + a 13 ⋅ b 32 [ a 11 a 12 a 13 ] [ b 12 b 22 b 32 ] = a 11 ⋅ b 12 + a 12 ⋅ b 22 + a 13 ⋅ b 32
• To obtain the entry in row 1, column 3 of A B , A B , multiply the first row of A A by the third column in B , B , and add. [ a 11 a 12 a 13 ] [ b 13 b 23 b 33 ] = a 11 ⋅ b 13 + a 12 ⋅ b 23 + a 13 ⋅ b 33 [ a 11 a 12 a 13 ] [ b 13 b 23 b 33 ] = a 11 ⋅ b 13 + a 12 ⋅ b 23 + a 13 ⋅ b 33

We proceed the same way to obtain the second row of A B . A B . In other words, row 2 of A A times column 1 of B ; B ; row 2 of A A times column 2 of B ; B ; row 2 of A A times column 3 of B . B . When complete, the product matrix will be

Properties of Matrix Multiplication

For the matrices A , B , A , B , and C C the following properties hold.

• Matrix multiplication is associative: ( A B ) C = A ( B C ) . ( A B ) C = A ( B C ) .
• Matrix multiplication is distributive: C ( A + B ) = C A + C B , ( A + B ) C = A C + B C . C ( A + B ) = C A + C B , ( A + B ) C = A C + B C .

Note that matrix multiplication is not commutative.

Multiplying Two Matrices

Multiply matrix A A and matrix B . B .

First, we check the dimensions of the matrices. Matrix A A has dimensions 2 × 2 2 × 2 and matrix B B has dimensions 2 × 2. 2 × 2. The inner dimensions are the same so we can perform the multiplication. The product will have the dimensions 2 × 2. 2 × 2.

We perform the operations outlined previously.

• ⓐ Find A B . A B .
• ⓑ Find B A . B A .
• ⓐ As the dimensions of A A are 2 × 3 2 × 3 and the dimensions of B B are 3 × 2 , 3 × 2 , these matrices can be multiplied together because the number of columns in A A matches the number of rows in B . B . The resulting product will be a 2 × 2 2 × 2 matrix, the number of rows in A A by the number of columns in B . B . A B = [ −1 2 3 4 0 5 ]    [ 5 −1 − 4 0 2 3 ] = [ −1 ( 5 ) + 2 ( −4 ) + 3 ( 2 ) −1 ( −1 ) + 2 ( 0 ) + 3 ( 3 ) 4 ( 5 ) + 0 ( −4 ) + 5 ( 2 ) 4 ( −1 ) + 0 ( 0 ) + 5 ( 3 ) ] = [ −7 10 30 11 ] A B = [ −1 2 3 4 0 5 ]    [ 5 −1 − 4 0 2 3 ] = [ −1 ( 5 ) + 2 ( −4 ) + 3 ( 2 ) −1 ( −1 ) + 2 ( 0 ) + 3 ( 3 ) 4 ( 5 ) + 0 ( −4 ) + 5 ( 2 ) 4 ( −1 ) + 0 ( 0 ) + 5 ( 3 ) ] = [ −7 10 30 11 ]
• ⓑ The dimensions of B B are 3 × 2 3 × 2 and the dimensions of A A are 2 × 3. 2 × 3. The inner dimensions match so the product is defined and will be a 3 × 3 3 × 3 matrix. B A = [ 5 −1 −4 0 2 3 ]    [ −1 2 3 4 0 5 ] = [ 5 ( −1 ) + −1 ( 4 ) 5 ( 2 ) + −1 ( 0 ) 5 ( 3 ) + −1 ( 5 ) −4 ( −1 ) + 0 ( 4 ) −4 ( 2 ) + 0 ( 0 ) −4 ( 3 ) + 0 ( 5 ) 2 ( −1 ) + 3 ( 4 ) 2 ( 2 ) + 3 ( 0 ) 2 ( 3 ) + 3 ( 5 ) ] = [ −9 10 10 4 −8 −12 10 4 21 ] B A = [ 5 −1 −4 0 2 3 ]    [ −1 2 3 4 0 5 ] = [ 5 ( −1 ) + −1 ( 4 ) 5 ( 2 ) + −1 ( 0 ) 5 ( 3 ) + −1 ( 5 ) −4 ( −1 ) + 0 ( 4 ) −4 ( 2 ) + 0 ( 0 ) −4 ( 3 ) + 0 ( 5 ) 2 ( −1 ) + 3 ( 4 ) 2 ( 2 ) + 3 ( 0 ) 2 ( 3 ) + 3 ( 5 ) ] = [ −9 10 10 4 −8 −12 10 4 21 ]

Notice that the products A B A B and B A B A are not equal.

This illustrates the fact that matrix multiplication is not commutative.

Is it possible for AB to be defined but not BA ?

Yes, consider a matrix A with dimension 3 × 4 3 × 4 and matrix B with dimension 4 × 2. 4 × 2. For the product AB the inner dimensions are 4 and the product is defined, but for the product BA the inner dimensions are 2 and 3 so the product is undefined.

Using Matrices in Real-World Problems

Let’s return to the problem presented at the opening of this section. We have Table 3 , representing the equipment needs of two soccer teams.

We are also given the prices of the equipment, as shown in Table 4 .

We will convert the data to matrices. Thus, the equipment need matrix is written as

The cost matrix is written as

We perform matrix multiplication to obtain costs for the equipment.

The total cost for equipment for the Wildcats is $2,520, and the total cost for equipment for the Mud Cats is $3,840.

Given a matrix operation, evaluate using a calculator.

• Save each matrix as a matrix variable [ A ] , [ B ] , [ C ] , ... [ A ] , [ B ] , [ C ] , ...
• Enter the operation into the calculator, calling up each matrix variable as needed.
• If the operation is defined, the calculator will present the solution matrix; if the operation is undefined, it will display an error message.

Using a Calculator to Perform Matrix Operations

Find A B − C A B − C given

On the matrix page of the calculator, we enter matrix A A above as the matrix variable [ A ] , [ A ] , matrix B B above as the matrix variable [ B ] , [ B ] , and matrix C C above as the matrix variable [ C ] . [ C ] .

On the home screen of the calculator, we type in the problem and call up each matrix variable as needed.

The calculator gives us the following matrix.

Access these online resources for additional instruction and practice with matrices and matrix operations.

• Dimensions of a Matrix
• Matrix Addition and Subtraction
• Matrix Operations
• Matrix Multiplication

7.5 Section Exercises

Can we add any two matrices together? If so, explain why; if not, explain why not and give an example of two matrices that cannot be added together.

Can we multiply any column matrix by any row matrix? Explain why or why not.

Can both the products A B A B and B A B A be defined? If so, explain how; if not, explain why.

Can any two matrices of the same size be multiplied? If so, explain why, and if not, explain why not and give an example of two matrices of the same size that cannot be multiplied together.

Does matrix multiplication commute? That is, does A B = B A ? A B = B A ? If so, prove why it does. If not, explain why it does not.

For the following exercises, use the matrices below and perform the matrix addition or subtraction. Indicate if the operation is undefined.

A + B A + B

C + D C + D

A + C A + C

B − E B − E

C + F C + F

D − B D − B

For the following exercises, use the matrices below to perform scalar multiplication.

1 2 C 1 2 C

100 D 100 D

For the following exercises, use the matrices below to perform matrix multiplication.

For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed.

A + B − C A + B − C

4 A + 5 D 4 A + 5 D

2 C + B 2 C + B

3 D + 4 E 3 D + 4 E

C −0.5 D C −0.5 D

100 D −10 E 100 D −10 E

For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. (Hint: A 2 = A ⋅ A A 2 = A ⋅ A )

B 2 A 2 B 2 A 2

A 2 B 2 A 2 B 2

( A B ) 2 ( A B ) 2

( B A ) 2 ( B A ) 2

( A B ) C ( A B ) C

A ( B C ) A ( B C )

For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. Use a calculator to verify your solution.

A B C A B C

For the following exercises, use the matrix below to perform the indicated operation on the given matrix.

Using the above questions, find a formula for B n . B n . Test the formula for B 201 B 201 and B 202 , B 202 , using a calculator.

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• IIT JEE Study Material
• Types Of Matrices
• Types of Matrices

There are different types of matrices, and they are basically categorised on the basis of the value of their elements, their order, the number of rows and columns, etc. Now, using different conditions, the various matrix types are categorised below, along with their definition and examples.

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All Contents in Matrices

• Introduction to Matrices
• Matrix Operations
• Adjoint and Inverse of a Matrix
• Rank of a Matrix and Special Matrices
• Solving Linear Equations Using Matrix

Download this lesson as PDF:- Types of Matrices PDF

JEE Main 2021 Maths LIVE Paper Solutions 24-Feb Shift-1 Memory-based

Matrix Types: Overview

The different types of matrices are given below:

Types of Matrices: Explanations

A matrix having only one row is called a row matrix . Thus A = [a ij ] mxn is a row matrix if m = 1. So, a row matrix can be represented as A = [a ij ] 1×n . It is called so because it has only one row, and the order of a row matrix will hence be 1 × n. For example, A = [1 2 4 5] is a row matrix of order 1 x 4. Another example of the row matrix is P = [ -4 -21 -17 ] which is of the order 1×3.

Column Matrix

A matrix having only one column is called a column matrix . Thus, A = [a ij ] mxn is a column matrix if n = 1. So, the value of a column matrix will be 1. Hence, the order is m × 1.

An example of a column matrix is:

Just like the row matrices had only one row, column matrices have only one column. Thus, the value of a column matrix will be 1. Hence, the order is m × 1. The general form of a column matrix is given by A = [a ij ] m×1 . Other examples of a column matrix include:

In the above example, P and Q are 3 ×1 and 5 × 1 order matrices, respectively.

Zero or Null Matrix

If all the elements are zero in a matrix, then it is called a zero matrix and generally denoted by 0. Thus, A = [a ij ] mxn is a zero-matrix if a ij = 0 for all i and j; E.g. \(\begin{array}{l}\left[ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{matrix} \right]\end{array} \) . is a zero matrix of order 2 x 3.

is a 3 x 2 null matrix and \(\begin{array}{l}B = \left[ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{matrix} \right]\end{array} \) is 3 x 3 null matrix.

Singleton Matrix

If there is only one element in a matrix, it is called a singleton matrix . Thus, A = [a ij ] mxn is a singleton matrix if m = n = 1. E.g. [2], [3], [a], [] are singleton matrices.

Horizontal Matrix

Vertical matrix, square matrix.

If the number of rows and the number of columns in a matrix are equal, then it is called a square matrix .

is a square matrix of order 3 × 3.

Another example of a square matrix is:

The order of P and Q is 2 ×2 and 3 × 3, respectively.

Diagonal Matrix

If all the elements, except the principal diagonal, in a square matrix, are zero, it is called a diagonal matrix . Thus, a square matrix A = [a ij ] is a diagonal matrix if a ij = 0,when i ≠ j.

(i) A diagonal matrix is always a square matrix

(ii) The diagonal elements are characterized by this general form: a ij where i = j. This means that a matrix can have only one diagonal.

A few more examples of a diagonal matrix are:

In the above examples, P, Q, and R are diagonal matrices with orders 1 × 1, 2 × 2 and 3 × 3, respectively. When all the diagonal elements of a diagonal matrix are the same, it goes by a different name, the scalar matrix, which is explained below.

Scalar Matrix

If all the elements in the diagonal of a diagonal matrix are equal, it is called a scalar matrix . Thus, a square matrix \(\begin{array}{l}A={{[{{a}_{ij}}]}_{m\times m}}\ \text{is a scalar matrix if}\ a_{ij}=\left\{ \begin{matrix} 0, & i\ne j \\ k, & i=j \\ \end{matrix}\right\}\end{array} \) where k is a constant.

More examples of scalar matrices are:

Now, what if all the diagonal elements are equal to 1? That will still be a scalar matrix and obviously a diagonal matrix. It has got a special name which is known as the identity matrix .

Unit Matrix or Identity Matrix

If all the elements of a principal diagonal in a diagonal matrix are 1, it is called a unit matrix . A unit matrix of order n is denoted by I n . Thus, a square matrix A = [a ij ] m×n is an identity matrix if

Conclusions:

• All identity matrices are scalar matrices
• All scalar matrices are diagonal matrices
• All diagonal matrices are square matrices

It should be noted that the converse of the above statements is not true for any of the cases.

Equal Matrices

Equal matrices are those matrices which are equal in terms of their elements. The conditions for matrix equality are discussed below.

• Equality of Matrices Conditions

Two matrices A and B are said to be equal if they are of the same order and their corresponding elements are equal, i.e. two matrices A = [a ij ] m×n and B = [b ij ] r×s are equal if:

(a) m = r, i.e., the number of rows in A = the number of rows in B.

(b) n = s, i.e. the number of columns in A = the number of columns in B

(c) a ij = b ij , for i = 1, 2, ….., m and j = 1, 2, ….., n, i.e. the corresponding elements are equal;

For example, Matrices \(\begin{array}{l}\begin{bmatrix} 0 &0 \\ 0& 0 \end{bmatrix} and \begin{bmatrix} 0 &0 & 0\\ 0& 0 &0 \end{bmatrix}\end{array} \) . are not equal because their orders are not the same.

But, If \(\begin{array}{l}A = \begin{bmatrix} 1 &6 &3\\ 5& 2&1 \end{bmatrix} and \begin{bmatrix} a_1&a_2 & a_3\\ b_1& b_2 &b_3 \end{bmatrix}\end{array} \) are equal matrices then,

a 1 = 1, a 2 = 6, a 3 = 3, b 1 = 5, b 2 = 2, b 3 = 1.

Triangular Matrix

A square matrix is said to be a triangular matrix if the elements above or below the principal diagonal are zero, and there are of two types:

• Upper Triangular Matrix

A square matrix [a ij ] is called an upper triangular matrix , if a ij = 0, when i > j.

• Lower Triangular Matrix

A square matrix is called a lower triangular matrix , if a ij = 0 when i < j.

Singular Matrix and Non-Singular Matrix

Matrix A is said to be a singular matrix if it’s determinant |A| = 0; otherwise, a non-singular matrix , i.e. if for det |A| = 0, it is singular matrix and for det |A| ≠ 0, it is non-singular.

Symmetric and Skew Symmetric Matrices

• Symmetric matrix: A square matrix A = [a ij ] is called a symmetric matrix if a ij = a ji , for all i,j values;

is symmetric, because a 12 = 2 = a 21 , a 31 = 3 = a 13 etc.

Note: A is symmetric if A’ = A (where ‘A’ is the transpose of the matrix)

• Skew-Symmetric Matrix: A square matrix A = [a ij ] is a skew-symmetric matrix if a ij = a ji , for all values of i,j.

are skew-symmetric matrices.

Note: A square matrix A is a skew-symmetric matrix A’ = -A.

Some Important Conclusions on Symmetric and Skew-Symmetric Matrices

• If A is any square matrix, then A + A’ is a symmetric matrix and A – A’ is a skew-symmetric matrix.
• Every square matrix can be uniquely expressed as the sum of a symmetric matrix and a skew-symmetric matrix. \(\begin{array}{l}A=\frac{1}{2}(A+A’)+\frac{1}{2}(A-A’)=\frac{1}{2}(B+C),\end{array} \) where B is symmetric, and C is a skew-symmetric matrix.
• If a and B are symmetric matrices, then AB is symmetric AB = BA, i.e., A & B commute.
• The matrix B’AB is symmetric or skew-symmetric in correspondence if A is symmetric or skew-symmetric.
• All positive integral powers of a symmetric matrix are symmetric.
• Positive odd integral powers of a skew-symmetric matrix are skew-symmetric, and positive even integral powers of a skew-symmetric matrix are symmetric.

Hermitian and Skew-Hermitian Matrices

A square matrix A = [a ij ] is said to be a Hermitian matrix if \(\begin{array}{l}{{a}_{ij}}={{\overline{a}}_{ji}}\,\forall \,i,j;\,i.e.\,A={{A}^{\theta }}\end{array} \) \(\begin{array}{l}E.g.\left[ \begin{matrix} a & b+ic \\ b-ic & d \\ \end{matrix} \right].\left[ \begin{matrix} 3 & 3-4i & 5+2i \\ 3+4i & 5 & -2+i \\ 5-2i & -2-i & 2 \\ \end{matrix} \right]\end{array} \) are Hermitian matrices

Important Notes:

• If A is a Hermitian matrix, then \(\begin{array}{l}{{a}_{ii}}={{\overline{a}}_{ii}}\Rightarrow {{a}_{ii}}\,is\,real\,\forall \,i,\end{array} \) thus every diagonal element of a Hermitian Matrix must be real.
• If a Hermitian matrix over the set of real numbers is actually a real symmetric matrix; and A a square matrix, A = [a ij ] is said to be a skew-Hermitian if \(\begin{array}{l}{{a}_{ij}}=-{{\overline{a}}_{ji}},\,\forall \,i,j;\end{array} \)

i.e., A θ = – A;

are skew-Hermitian matrices.

• If A is a skew-Hermitian matrix then \(\begin{array}{l}{{a}_{ii}}=-{{\overline{a}}_{ii}}\Rightarrow a_{ii}+\overline{a_{ii}}=0\end{array} \)

i.e., a ii must be purely imaginary or zero.

• A skew-Hermitian matrix over the set of real numbers is actually a real skew-symmetric matrix.

Special Matrices

(a) Idempotent Matrix:

For an idempotent matrix A, det A = 0 or x.

(b) Nilpotent Matrix:

A nilpotent matrix is said to be nilpotent of index p, \(\begin{array}{l}\left( p\in N \right),\;\; if \;\;{{A}^{p}}=O,\,\,{{A}^{p-1}}\ne O,\end{array} \) , i.e. if p is the least positive integer for which A p = O, then A is said to be nilpotent of index p.

(c) Periodic Matrix:

A square matrix which satisfies the relation A k + 1 = A, for some positive integer K, then A is periodic with period K, i.e. if K is the least positive integer for which A k + 1 = A, and A is said to be periodic with period K. If K =1, then A is called idempotent.

has period 1.

(i) Period of a square null matrix is not defined.

(ii) Period of an idempotent matrix is 1.

(d) Involutory Matrix:

If A 2 = I, the matrix is said to be an involutory matrix. An involutory matrix with its own inverse.

Matrices and Determinants – Important Topics

Matrices and Determinants – Important Questions

Frequently Asked Questions

What do you mean by a symmetric matrix.

A symmetric matrix is defined as a square matrix which is equal to its transpose matrix. We denote the transpose of a matrix A as A T . A symmetric matrix A will satisfy the condition A T = A.

What do you mean by identity matrix?

An identity matrix is a square matrix whose diagonal elements are 1, and other elements are zero. The identity matrix is also called the unit matrix.

What do you mean by Hermitian matrix?

If a square matrix is equal to its conjugate transpose matrix, then the matrix is called a Hermitian matrix.

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Dodgers lefty James Paxton designated for assignment as River Ryan makes his major league debut

By joe reedy, associated press | posted - july 22, 2024 at 7:49 p.m..

Estimated read time: Less than a minute

LOS ANGELES — The Los Angeles Dodgers have made a surprising change to their evolving rotation. James Paxton was designated for assignment to make room for River Ryan on the 40-man roster. Ryan made his major league debut Monday night against San Francisco, becoming the 14th pitcher to start for the Dodgers this season. Paxton was tied with Gavin Stone and Tyler Glasnow for most starts on the team with 18. The veteran left-hander was 8-2 with a 4.43 ERA. The 35-year-old Paxton signed a $7 million, one-year contract during the offseason. He allowed two runs over five innings in Sunday's 9-6 victory against Boston.

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Dodgers lefty James Paxton designated for assignment as River Ryan makes his major league debut

Los Angeles Dodgers starting pitcher James Paxton throws during the first inning of a baseball game against the Boston Red Sox, Sunday, July 21, 2024, in Los Angeles. (AP Photo/Ryan Sun)

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LOS ANGELES (AP) — The Los Angeles Dodgers made a surprising change to their evolving rotation Monday.

James Paxton was designated for assignment to make room for River Ryan on the 40-man roster. Ryan made his major league debut Monday night against the San Francisco Giants, becoming the 14th pitcher to start for the Dodgers this season.

Paxton was tied with Gavin Stone and Tyler Glasnow for most starts on the team with 18. The veteran left-hander was 8-2 with a 4.43 ERA.

The 35-year-old Paxton signed a $7 million, one-year contract during the offseason. He allowed two runs over five innings in Sunday’s 9-6 victory against the Boston Red Sox.

Los Angeles manager Dave Roberts said with the return of Glasnow and Clayton Kershaw from injuries this week along with some young arms the Dodgers want to look at, it was tough finding a fit for Paxton the rest of the way.

“It was a difficult decision. He handled it like a pro,” Roberts said. “We feel good about the starting staff going forward.”

Glasnow, who went on the injured list July 9 due to back tightness, will be activated and start on Wednesday. Kershaw is scheduled to make his first start of the season Thursday in the series finale after working his way back from offseason shoulder surgery.

Even with the returns of Glasnow and Kershaw, the Dodgers are still missing Walker Buehler (right hip inflammation) and Yoshinobu Yamamoto (right rotator cuff).

Buehler will throw a bullpen Tuesday before making a couple of minor league rehab starts. Yamamoto isn’t expected back until possibly late August.

Justin Wrobleski and Landon Knack are two of the younger pitchers the Dodgers want to see more. Knack will start Tuesday night.

With the July 30 deadline looming, the Dodgers are likely to find a trade partner for Paxton. Los Angeles has seven days to trade, release or send him outright to the minors — an assignment he would have the right to decline in favor of free agency.

Roberts’ bigger concern might be with the bullpen. Roberts indicated he is likely to go with a closer-by-committee approach due to Evan Phillips’ recent struggles.

Phillips has allowed 10 runs (nine earned) in his last eight appearances and has an 11.05 ERA since June 30. He gave up three runs in one-third of an inning on Sunday against Boston.

“For me, the command is just a little bit off. Over the past couple weeks, he was one pitch away from having a good outing,” Roberts said about Phillips. “We need him to be good, and it’s our job to get him back to being who he is.”

AP MLB: https://apnews.com/hub/mlb

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San Francisco Giants Set Official Date for Cy Young Winner's Team Debut

Brad wakai | 16 hours ago.

• San Francisco Giants

Right before the All-Star break, the San Francisco Giants started playing better baseball.

There is still time for them to make a push towards the playoffs if they can consistently perform at that level, but that has been the issue for this group all season long. Injuries were a huge factor during their disappointing first half, and with multiple impact players set to return, there's a chance they are one of the most improved teams heading into the fall.

Getting reigning NL Cy Young winner Blake Snell back and performing like the elite starting pitcher he is has been a huge sigh of relief.

Now, he won't be the only past Cy Young winner in this rotation as Alex Pavlovic of NBC Sports Bay Area reports Robbie Ray will officially make his Giants debut on Wednesday, July 24 against their archrival Los Angeles Dodgers .

Ray was acquired by San Francisco this past offseason in a trade with the Seattle Mariners that saw them ship out Mitch Haniger, Anthony DeSclafani, and cash considerations to acquire the star left-hander.

He only pitched in 33 games for the Mariners after signing a five-year, $115 million contract heading into the 2022 season. Last year, he underwent Tommy John surgery after one outing, which has kept him sidelined until this point.

The Giants decided to take a chance on the rehabbing starter who has a career 3.96 ERA and 11.0 K/9 ratio during his 10-year career.

Ray has made 10 rehab outings and nine starts as he's worked his way back into the mix with his new team. During this time, he's posted a 3.38 ERA by allowing 11 earned runs on 19 hits across 29.1 innings pitched.

Brad Wakai graduated from Penn State University with a degree in Journalism. While an undergrad, he did work at the student radio station covering different Penn State athletic programs like football, basketball, volleyball, soccer and other sports. Brad currently is the Lead Contributor for Nittany Lions Wire of Gannett Media where he continues to cover Penn State athletics. He is also a contributor at FanSided, writing about the Philadelphia 76ers for The Sixers Sense. Brad is the host of the sports podcast I Said What I Said, discussing topics across the NFL, College Football, the NBA and other sports. You can follow him on Twitter: @bwakai