Operations on Matrices: Addition, Subtraction, Multiplication and Solved Examples

A matrix can be defined as a rectangular array of expressions or numbers. The numbers or expressions present within a matrix are known as elements. The horizontal lines within a matrix are referred to as rows and the vertical lines are known as columns. Thus, a matrix with ‘a’ as rows and ‘b’ columns can be represented as a matrix of the order a * b.

Keyterms: Matrices, Row Matrix, Column Matrix, Square Matrix, Diagonal Matrix, Scalar Matrix, Identity Matrix and Zero Matrix

Read More: Minors and Cofactors

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Types of Matrices

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• Row Matrix - A matrix have only one row
• Column Matrix- A matrix constituting of only one column
• Square Matrix - A matrix containing equal number of rows and columns
• Diagonal Matrix - A square matrix whose non diagonal elements are equal to zero.
• Scalar Matrix - A square matrix whose diagonal elements are equal.
• Identity Matrix - A square matrix whose diagonal elements are equal to 1 and rest are zero.
• Zero Matrix - A matrix where all its elements are zero.

The video below explains this:

Matrices Detailed Video Explanation:

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Matrix operations

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The range of operations that can be performed on a matrix involves three algebraic operations mainly. These three operations are namely - 

1. Addition of matrices
2. Subtraction of matrices
3. Multiplication of matrices

The basic operations among these are the addition and subtraction to matrices. For these operations, the matrices involved in the operation must be of identical order. However while multiplying, the number of columns of the first matrix must be equal to the number of rows of the second matrix. Let us understand these operations in detail.

1. Addition of Matrices

Let us think that A[a_ij]_m*n and B[b_ij]_m*n are two matrices of the same order m * n. So, when we add these two matrices, each element of the resulting matrix will be a sum of [a_ij + b_ij]_m*n i.e. it will be a sum of the corresponding values of the A and B matrix, which can be represented as matrix C = [c_ij]_m*n. Hence

\(\begin{bmatrix}a1 & b1 \\[0.3em]c1 & d1 \\[0.3em] \end{bmatrix} + \begin{bmatrix}a2 & b2 \\[0.3em]c2 & d2 \\[0.3em] \end{bmatrix}= \begin{bmatrix}a1 + a2 & b1 + b2 \\[0.3em]c1+ c2 & d1+d2 \\[0.3em] \end{bmatrix}\)

Properties of Addition of Matrices

• Commutative Law - According to the commutative law of addition

If A = [a_ij], B = [b_ij] of the order m * n, then A+B = B+A

• Associative Law

If A = [a_ij], B = [b_ij], C = [c_ij] of the order m * n, then (A + B) + C = A + (B + C)

• Additive Inverse

If A = [a_ij] is a matrix of the order m * n, then its additive inverse -A = [- a_ij]_mn is developed by changing the sign of every element of A. Thus, we have: 

A + (-A) = 0 = (-A) + A

If A+B = 0 = B+A, then A and B can be called the additive inverse of each other.

tr(A+B) = tr(A) + tr(B),

Similarly, tr(A-B) = tr(A) - tr(B)

• Additive Identity of the Matrix

If A = [a_ij] is a matrix of the order m*n and O is a zero matrix of the m * n order, then

A + O = O + A = A

Here O is known as the additive identity of the matrix.

2. Subtraction of Matrices

If there are two matrices namely A and B and they both are of the same order m * n., then the difference of these two matrices A - B can be written as D = [d_ij]. Here d_ij = a_ij - b_ij. We can also say that D = A+(-1) B.

The new elements of the resulting matrix can be found by calculating the difference between the corresponding elements of both the first and second matrix. Thus, 

\(\begin{bmatrix}a1 & b1 \\[0.3em]c1 & d1 \\[0.3em] \end{bmatrix} - \begin{bmatrix}a2 & b2 \\[0.3em]c2 & d2 \\[0.3em] \end{bmatrix}= \begin{bmatrix}a1 - a2 & b1 - b2 \\[0.3em]c1- c2 & d1-d2 \\[0.3em] \end{bmatrix}\)

3. Multiplication of Matrices

Two matrices A and B can be multiplied only when the number of columns of the ‘A’ matrix is equal to the number of rows of the ‘B’ matrix.

i.e. let us consider A[a_ij]m*n and B[b_ij]n*p. Then the product of these two matrices can be written as C[C_ij]m*p.

Properties of Multiplication of Matrices

• The multiplication of these matrices A and B is not commutative in nature. This means AB is not equal to BA.
• Matrix multiplication is associative. This means that for three given Matrices A, B, C we will have (AB)C = A(BC)
• Matrix multiplication also follows the property of distributive law. This law states if three matrices A,B and C are given, then - 
  • A(B+C) = AB + AC
  • (A + B) C = AC + BC [In this case both sides of the equality must be defined]
  • An identity matrix of the same order will be present for every square matrix ‘A’. Thus, we can say IA = AI = A.

Scalar Multiplication of Matrices

While dealing with Matrices, any real number is called scalar. Thus, in case of scalar multiplication, a real number is multiplied with a matrix. Therefore, If k is any real number and A is a matrix which is [a_ij]m*n then the multiplication of k with A matrix is the scalar multiplication and can be written as: 

 kA=k[a_ij]_m*n =[k(a_i×j)]_{m * n}

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Solved Examples

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Ques. If A =\(\begin{bmatrix}5 & 6 \\[0.3em]1 & 2 \\[0.3em] \end{bmatrix}\) and B = \(\begin{bmatrix}2 & 9 \\[0.3em]7 & 8 \\[0.3em] \end{bmatrix}\) then what is A + B?  (2 marks)

Ans. As A and B are two matrices of the same order, we can get the matrix A+B by adding the corresponding elements of A and B.

So, A + B = \(\begin{bmatrix}7 & 15 \\[0.3em]8 & 10 \\[0.3em] \end{bmatrix}\)

Ques. Demonstrate A+B is commutative if A =  \(\begin{bmatrix}5 & 5 \\[0.3em]3 & 6 \\[0.3em] \end{bmatrix}\)and B = \(\begin{bmatrix}2 & 9 \\[0.3em]5 & 1 \\[0.3em] \end{bmatrix}\) (2 marks)

Ans. If A and B are two matrices of the same order, then A+B is commutative if A+B = B+A. It can be understood as we know that if A and B are of the same order, then the elements of the resulting matrix A + B is equal to the addition of corresponding elements of A and B.

We can see this is true by actually performing the calculations

So, \(\begin{bmatrix}5 & 5 \\[0.3em]3 & 6 \\[0.3em] \end{bmatrix}\) + \(\begin{bmatrix}2 & 9 \\[0.3em]5 & 1 \\[0.3em] \end{bmatrix}\) = \(\begin{bmatrix}2 & 9 \\[0.3em]5 & 1 \\[0.3em] \end{bmatrix}\) + \(\begin{bmatrix}5 & 5 \\[0.3em]3 & 6 \\[0.3em] \end{bmatrix}\) = \(\begin{bmatrix}7 & 14 \\[0.3em]8 & 7 \\[0.3em] \end{bmatrix}\)

Hence A + B is commutative in nature if A and B are two matrices of the same order.

Read Also: Properties of Determinants

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Things to Remember 

• A matrix can be defined as a rectangular array of expressions or numbers.
• The numbers or expressions present within a matrix are known as elements.
• The horizontal lines within a matrix are referred to as rows and the vertical lines are known as columns.
• A matrix with ‘a’ as rows and ‘b’ columns can be represented as a matrix of the order a * b.

• Types of Matrices are Row Matrix, Column Matrix, Square Matrix, Diagonal Matrix, Scalar Matrix, Identity Matrix and Zero Matrix.