## 1. Concept of Matrices and Types of Matrices

Chapter-3

Matrices

Concept of Matrices and Types of Matrices

Matrices Definition

A matrix is a rectangular array of numbers or functions or objects  arranged in row and column.

The numbers or functions or objects in the array are called the entities or members or the elements of the matrix. The horizontal array of elements in the matrix is called rows, and the vertical array of elements are called the columns. If a matrix has m rows and n columns, then it is known as the matrix of order ‘m x n’ or ‘m by n’.

Symbol: ( ) or [ ]

i.e., A = [aij]m×n  means m rows and n columns. Note: If A = [aij] n×n  is a square matrix of order n, then elements a11, a22, a33,…, ann is said to constitute the diagonal of the matrix A. In general, A = [aij]m×m is a diagonal matrix, if aij = 0, when i ≠ j.

## Types of Matrices:

1. Column matrix
2. Row matrix
3. Square matrix
4. Diagonal matrix
5. Scalar matrix
6. Identity matrix
7. Zero matrix   Equality of Matrices

Two matrices are said to be equal if-

(i) The order of both the matrices are the same

(ii) Each element of one matrix is equal to the corresponding element of the other matrix

i.e., ## Operations on Matrices

The  addition of matrices, multiplication of a matrix by a scalar, difference and multiplication of matrices. However, division of matrices is not possible in a direct way.

If   A=[aij] and   B=[bij] are two matrices of the same order, say m × n, then the sum of the two matrices A and B is defined as a matrix C=[cij]m×n , where cij=aij+bij , for all possible values of i and j.

i.e. C = A + B are two matrices of order 3 x 2 such that the sum of these two matrices is given by:

C = A + B

Therefore, That means the sum or addition of two matrices is a matrix obtained by adding the corresponding elements of the given two matrices. Also, it is essential to note that the two matrices have to be of the same order. Here, the order of the matrix A is 2 x 3 and the order of B is 2 x 3 are the same. So, we can add the given two matrices by adding the corresponding elements. Example: Suppose , then the addition of A and B is not possible since the order of matrix A is 2 x 2 and the order of B is 2 x 3, i.e. the order of these matrices is not equal.

## Properties of Addition of Matrices

 Commutative Law: If A = [aij], B = [bij] are matrices of the same order, say m × n, then:

A + B = B + A

i.e. A + B = [aij] + [bij]

= [aij + bij]

= [bij + aij] (addition of numbers is commutative)

= ([bij] + [aij])

= B + A

Therefore, the addition of matrices is commutative.

 Associative Law: For any three matrices A = [aij], B = [bij], C = [cij] of the same order, say m × n, then:

(A + B) + C = A + (B + C)

This can be shown as:

(A + B) + C = ([aij] + [bij]) + [cij]

= [aij + bij] + [cij]

= [(aij + bij) + cij]

= [aij + (bij + cij)] (addition of numbers is associative)

= [aij] + [(bij + cij)]

= [aij] + ([bij] + [cij])

= A + (B + C)

 Existence of additive identity: Let A = [aij] be an m × n matrix and O be an m × n zero matrix, then:

A + O = O + A = A

 The existence of additive inverse: Let A = [aij] be any matrix of the order m × n, then we have another matrix as –A = [–aij]m × n such that A + (–A) = (–A) + A = O.

Thus, –A is the additive inverse of A or negative of A. The negative of a matrix is denoted by –A and it can be defined as –A = (–1) A. Now, by equating the corresponding elements,

2x + 2 = 8

2x = 6

x = 3

Also, 2 + y = 5

y = 3

Therefore, x = 3 and y = 3.

Subtraction of matrices:

The Subtraction of matrices is possible only when the order of the two matrices is the same.

but for multiplication of matrices, we need to check if the number of columns of one matrix is equal to the number of rows of the second matrix.

If there are two matrices, say A = [aij] and B = [bij] of the same order, say m × n, then the subtraction of A and B, i.e., A – B is defined as:

Matrix D = [dij]

A – B = aij – bij

Thus,

dij = aij – bij, (i = 1,2,3,… and j= 1,2,3…)

D = A – B = aij – bij

A – B = A + (-B)

### Subtraction of 2 x 2 Matrices

Suppose A and B are 2 x 2 matrices, such that; Then, subtraction of matrices A and B, will be given as: Fact: If A and B are two matrices of the same order, then;

A – B ≠ B – A

Thus, commutative law is not applicable for subtraction of matrices.

### Subtraction of 3 x 3 Matrices

Suppose A and B are 3 x 3 matrices, such that; Then, subtraction of matrices A and B, will be given as: ## Examples

Q.1: If A  and B are two matrices. Then find subtraction of matrices A and B.  ## Matrix Multiplication Definition

Matrix multiplication, also known as matrix product and the multiplication of two matrices, produces a single matrix. It is a type of binary operation.

If A and B are the two matrices, then the product of the two matrices A and B are denoted by:

X = AB

Hence, the product of two matrices is the dot product of the two matrices.

Matrix multiplication by Scalar

Multiplication of an integer with a matrix is simply a scalar multiplication.

We know that a matrix is an array of numbers. It consists of rows and columns. If you multiply a matrix by a scalar value, then it is known as scalar multiplication. Another case is that it is possible to multiply a matrix by another matrix. Let’s have a look at the example given below for the same.

We may define multiplication of a matrix by a scalar mathematically as:

If A = [aij]m × n is a matrix and k is a scalar, then kA is another matrix obtained by multiplying each element of A by the scalar k.

In other words, kA = k [aij]m × n = [k (aij)]m × n, that is, (i, j)th element of kA is kaij for all possible values of i and j.

Example: Multiply the matrix  Matrix multiplication Condition

To perform multiplication of two matrices, we should make sure that the number of columns in the 1st matrix is equal to the rows in the 2nd matrix. Therefore, the resulting matrix product will have a number of rows of the 1st matrix and a number of columns of the 2nd matrix. The order of the resulting matrix is the matrix multiplication order.

Four  Types of multiplication :

i) row by row

ii) row by column

iii) column by column

iv) column by row

Normally we multiply row by column method.

## How to Multiply Matrices?

Let’s learn how to multiply matrices.

Consider matrix A which is a × b matrix and matrix B, which is a b ×c matrix.

Then, matrix C = AB is defined as the A × B matrix.

An element in matrix C, Cxy is defined as Cxy = Ax1By1 +….. + AxbBby =

AxkBky  for x = 1…… a  and y= 1…….c

### Notation

If A is a m×n matrix and B is a p×q matrix, then the matrix product of A and B is represented by:

X = AB

Where X is the resulting matrix of m×q dimension.

Matrix Multiplication Formula

Let’s take an example to understand this formula.

Let’s say A and B are two matrices, such that, Then Matrix C = AB is denoted by An element in matrix C where C is the multiplication of Matrix A X B.

C = Cxy = Ax1By1 +….. + AxbBby =(AB)ij=Cij=r=1nairbrj AxkBky  for x = 1…… a  and y= 1…….c

### Matrix multiplication Rules

Rules and properties for matrix multiplication.

• The product of two matrices A and B is defined if the number of columns of A is equal to the number of rows of B.
• If AB is defined, then BA need not be defined
• If both A and B are square matrices of the same order, then both AB and BA are defined.
• If AB and BA are both defined, it is not necessary that AB = BA.
• If the product of two matrices is a zero matrix, it is not necessary that one of the matrices is a zero matrix.

## 2×2 Matrix Multiplication

Let a simple 2 × 2 matrix multiplication Let a simple 2 × 2 matrix multiplication And  another  matrix Now each of the elements of product matrix AB can be calculated as follows:

• AB11 = 3 × 6 + 7 ×5 = 53
• AB12 = 3 × 2 + 7 × 8 = 62
• AB21 = 4 × 6 + 9 × 5 = 69
• AB22 = 4 × 2 + 9 × 8 = 80 3×3 Matrix Multiplication  The multiplication of two 3 × 3 matrices, let two 3 × 3 matrices A and B. Each element of the Product matrix AB can be calculated as follows:

• AB11 = 12×5 + 8×6 + 4×7 = 136
• AB12 = 12×19 + 8×15 + 4×8 = 380
• AB13 = 12×3 + 8×9+4×16 = 172
• AB21 = 3×5 + 17×6 + 14×7 = 215
• AB22 = 3×19 + 17×15 + 14×8 = 424
• AB23 = 3×3 + 17×9 + 14×16 = 386
• AB31 = 9×5 + 8×6 + 10×7 = 163
• AB32 = 9×19 + 8×15 + 10×8 = 371
• AB33 = 9×3 + 8×9 + 10×16 = 259   ## Properties of Matrix Multiplication

The following are the properties of the matrix multiplication:

### Commutative Property

The matrix multiplication is not commutative.

Assume that, if A and B are the two 2×2 matrices,

AB ≠ BA

In matrix multiplication, the order matters a lot.

For example, This shows that the matrix AB ≠BA.

Hence, the multiplication of two matrices is not commutative.

### Associative Property

If A, B and C are the three matrices, the associative property of matrix multiplication states that,

(AB) C = A(BC)  Hence, the associative property of matrix multiplication is proved.

### Distributive Property

If A, B and C are the three matrices, the distributive property of matrix multiplication states that,

• (B+C)A = BA +CA
• A(B+C) = AB + AC

### Multiplicative Identity Property

The identity property of matrix multiplication states that,  ### Dimension Property

In matrix multiplication, the product of m × n matrix and n×a matrix is the m× a matrix.

For example, matrix A is a 2 × 3 matrix and matrix B is a 3 × 4 matrix, then AB is a 2 × 4 matrices.

### Multiplicative property of Zero

If a matrix is multiplied by a zero matrix, the result matrix is a zero matrix.

## 3. Transpose of a matrices and Properties of transpose of the matrices

### Transpose of a Matrix

If A = [aij] be an m × n matrix, then the matrix obtained by interchanging the rows and columns of A is called the transpose of A and is denoted by A′ or (AT ).

In other words, if A = [aij] m × n , then A′ = [aji] n × m .

Example: Properties of Transpose of a Matrix ## 4. Symmetric and Skew Symmetric Matrices

### Symmetric and Skew Symmetric Matrices

A square matrix A = [aij] is said to be symmetric if the transpose of A is equal to A, that is, [aij] = [aji] for all possible values of i and j.

A square matrix A = [aij] is a skew-symmetric matrix if A′ = – A, that is aji = – aij for all possible values of i and j. Also, if we substitute i = j, we have aii = – aii and thus, 2aii = 0 or aii = 0 for all i’s. Therefore, all the diagonal elements of a skew symmetric matrix are zero.

Properties of Symmetric Matrix

• Addition and difference of two symmetric matrices results in symmetric matrix.
• If A and B are two symmetric matrices and they follow the commutative property,

i.e. AB =BA, then the product of A and B is symmetric.

• If matrix A is symmetric then An is also symmetric, where n is an integer.
• If A is a symmetrix matrix then A-1 is also symmetric.

### Properties of Skew Symmetric Matrix

• When we add two skew-symmetric matrices then the resultant matrix is also skew-symmetric.
• Scalar product of skew-symmetric matrix is also a skew-symmetric matrix.
• The diagonal of skew symmetric matrix consists of zero elements and therefore the sum of elements in the main diagonals is equal to zero.
• When identity matrix is added to skew symmetric matrix then the resultant matrix is invertible.
• The determinant of skew symmetric matrix is non-negative

Theorem :  How do you prove symmetric and skew symmetric matrix? Example Properties of Symmetric and Skew-Symmetric Matrices

There are some rules that come from the concept of Symmetric and Skew-Symmetric Matrices,

1. If matrix A is a square matrix then (A + At) is always symmetric.

Prove:

To find if a matrix symmetric or not, first, we have to find the transposed form of the given matrix

So, let’s find the transpose of (A + At)

= (A + At)t

= A+ (At)t

= A+ A   [here, (At)= A]

= (A + At)

So, this is the same as the given matrix, so it is symmetric.

2. If matrix A is a square matrix then (A – At) is always skew-symmetric.

Prove:

To find if a matrix skew-symmetric or not, first, we have to find the transposed form of the given matrix

So, let’s find the transpose of (A – At)

= (A − At)t

= A− (At)t

= A− A  [here, (At)t = A]

= − (A − At

So, this form is the negative of the given matrix, so it is skew-symmetric.

Theorem 1:

Any Square matrix can be expressed as the sum of a symmetric and a skew-symmetric matrix. Proof: Let A be a square matrix then, we can write A = 1/2 (A + A′) + 1/2 (A − A′). From the Theorem 1, we know that (A + A′) is a symmetric matrix and (A – A′) is a skew-symmetric matrix.

Theorem 2: Any square matrix can be expressed as the sum of a symmetric and a skew symmetric matrix.

Proof: Let A = [aij]mxm

We know that A + A' : Symmetric matrix

A - A' : Skew Symmetric matrix

½(A + A' ) + ½ (A - A')

= (½A + ½A') + (½A - ½A') Using kA = k(A)

½A + ½A = A.

Theorem 3: Inverse of matrix A is unique.

Proof:

Let B and C be two inverses of A.

i.e., AxB=BxA=I

and  AxC =CxA=1

Now, B=BI = B (AxC)  = ( BxA) X C      = I xC    =     C

So, inverses are unique.

## 5. Invertible Matrices and Inverse of a matrix by elementary operations

Invertible Matrices and Inverse of a matrix by elementary operations

Elementary Operation (Transformation) of a Matrix

There are six operations (transformations) on a matrix, three of which are due to rows, and three are due to columns, known as elementary operations or transformations.

 The interchange of any two rows or two columns.

denoted by Ri ↔ Rj and interchange of i th and j th column is denoted by Ci ↔ Cj .

 The multiplication of the elements of any row or column by a non zero number.

denoted by Ri → kRi . The corresponding column operation is denoted by Ci → kCi

 The addition to the elements of any row or column, the corresponding elements of any other row or column are multiplied by any non zero number.

denoted by Ri → Ri + kRj . The corresponding column operation is denoted by Ci → Ci + kCj .

Elementary transformations are operations done on the rows and columns of matrices to change their shape so that the computations become easier. It is also used to discover the inverse of a matrix, the determinants of a matrix, and to solve a system of linear equations. A square matrix is always an elementary matrix.

Invertible Matrices

Suppose a square matrix A of order m, and if there exists another square matrix B of the same order m, such that AB = BA = I, then B is called the inverse matrix of A, and it is denoted by A-1. Also, matrix A is said to be an invertible matrix here.

Note:-

 A rectangular matrix does not possess inverse matrix, since for products BA and AB to be defined and to be equal, it is necessary that matrices A and B should be square matrices of the same order.

 If B is the inverse of A, then A is also the inverse of B.

Theorem 4:  If A and B are invertible matrices of the same order,

then (AB)–1 = B–1 A–1

Proof : From the definition of inverse of a matrix,

we have (AB) (AB)–1 = I

or A–1 (AB) (AB)–1 =A–1I (Pre multiplying both sides by A–1)

or (A–1A) B (AB)–1 =A–1 (Since A–1 I = A–1)

or IB (AB) –1  =A–1  or B (AB) –1  =A–1

or B–1  B (AB) –1  =B–1 A–1

or I (AB) –1  =B–1  A–1

Hence (AB) –1  =B–1  A–1

Example 1: , then find the value of a, b, c, x, y, and z.

Solution:

It is given that, the two matrices are equal. Therefore, the corresponding elements present in matrices should be equal to each other. By comparing the corresponding elements in the matrices, we get:

x+3 = 0. ⇒ x = -3

z +4 = 6  ⇒ z = 6-4

⇒ z = 2

2y-7 = 3y-2     ⇒3y-2y =-7+2

⇒y = -5

a-1 = -3

⇒a = -3+1

⇒a=-2

2c+2 = 0

⇒2c = -2

⇒ c = -1

b-3 = 2b+4

⇒2b-b = -3-4

⇒ b = -7

Therefore, the values of the variables are:

a = -2

b = -7

c = -1

x = -3

y = -5

z = 2

Example 2: Now, we need to calculate the transpose of AB. Hence  verified.