Matrix multiplication,Properties,Practice Problems and FAQs

Multiplication as we know in algebra is a binary operation which produces a number form two numbers. Similarly this operation can be performed in case of matrices also. Here, matrix multiplication is an operation that produces a matrix from two matrices.

Matrix multiplication is probably the most important operation on matrices which is used widely in areas such as transformation of coordinate systems, solution of linear system of equations , network theory and population modeling etc.Let us know try to dig into the method for matrix multiplication & it’s properties in details.

Table of contents:

Matrix multiplication
Process of Matrix multiplication
Properties of matrix multiplication
Practice Problems
FAQs

Matrix multiplication

Product of two matrices A & B will exist only when the number of columns of the first matrix i.e. A is the same as the number of rows of the second matrix i.e. B

$A = {[a_{i j}]}_{m \times n} a n d B = {[b_{i j}]}_{n \times p} A_{m \times n} B_{n \times p} = C_{m \times p} = {[c_{i k}]}_{m \times p} w h e r e c_{i k} = \sum_{j = 1}^{n} a_{i j} \times b_{j k}$

Process of Matrix Multiplication

In matrix multiplication for obtaining the ij^th element of the resultant matrix multiply the corresponding elements of the i^th row of the first matrix with the j^th column of the second matrix and add the products.

Let us look at an example

∵ Number of columns in A is equal to number of rows in B ,so multiplication is possible.

Note:

In product of AB ,A is called pre multiplier and B is called post multiplier.
If matrix multiplication if AB is defined that doesn’t mean BA is also defined for matrices A and B .
In multiplication, the sequence in which the matrices are written in the product matters.

Properties of matrix multiplication

Commutative Property: In general matrix multiplication is not commutative, even if both AB & BA are defined i.e. ABBA.

Associative Property: Matrix multiplication is associative. If A, B & C are three matrices such that A(BC) & (AB)C both are defined, then A(BC)=(AB)C

Distributive Property: Matrix multiplication is distributive over addition. For three matrices A,B & C , we have A(B+C)=AB+AC

Multiplicative Identity Property: The identity property of matrix multiplication states that

A.I=I.A=A where A & I are square matrices of the same order.

Note:

(i) Positive integral powers of a square matrix: For a square matrix A of order n ,

A²=A.A

A³=A.A.A=A.A²=A².A

A⁴=A.A.A.A=A².A²=A.A³=A³.A

and so on.

(ii) (A+B)²=(A+B).(A+B)

= A²+A.B+B.A+B²

(A-B)²=(A-B).(A-B)

=A²-B.A-A.B-B²

Practice Problems:

Example:

Solution:

In such types of problem ,we find a particular pattern and get the desired matrix.

Now, simply by observing the pattern ,we will find An

Example: If A is a 3×2 matrix and B is a matrix such that both AB and BA are defined, then the order of B is

a) 2×3
b) 3×2
c) 3×3
d) 2×2

Solution:

Order of matrix A = 3×2. Let order of matrix B be p × q

For AB to be defined the number of columns in A should be equal to the number of rows in B Hence, p=2

For BA to be defined the number of columns in B should be equal to the number of rows in A

Hence, q=3

Therefore, Order of B=23

Example: Which of the following properties of matrix multiplication is correct?

a) Multiplication is distributive over addition
b) Multiplication is not commutative in general
c) Multiplication is associative
d) All of these

Solution:

As we know that matrix multiplication is distributive over addition,matrix multiplication is associative and matrix multiplication is non commutative.

Hence, option (d) is correct.

Example: Let be a root of the equation x² +x+1=0 and a matrix A is given by

, then the matrix A³¹ is equal to.

a) A
b) A³
c) A²
d) I₃

Solution:

The roots of the equation x² +x+1=0 are

Hence,a is complex cube root of unity

Take 3 common from the matrix,

Thus the correct option is (b)

Example: If A & B are two square matrices such that they commute then show that n N,

ABⁿ=BⁿA .

Solution:

L.H.S =A.Bⁿ

=A.B.B^n-1

=B.A.B^n-1

=B.A.B.B^n-2

=B.B.A.B^n-2 ( A.B=B.A as A & B commmute )

=B².A.B.B^n-3

=B³.A.B.B^n-4 and so on…..

=Bⁿ.A

Hence, A.Bⁿ=Bⁿ.A

FAQs

Q 1. Can division of matrices be possible?
Answer: There is no such thing as division in matrices. Matrices can be added, subtracted, and multiplied, but they cannot be divided.

Q 2. Is pre and post multiplication the same?
Answer: No, it is not the same.The multiplier is on the left when pre-multiplying a matrix by another matrix. It's on the right when we perform post multiplication.

Q 3. What is the result if we multiply a matrix with its inverse?
Answer: If we multiply a matrix with its inverse we get an identity matrix of the same order as the given matrix.

Q 4. Is the multiplication of 3×4 & 4×5 matrices possible?
Answer: Yes, the product of these two matrices is possible as the number of columns in the first matrix and the number of rows in the second matrix are equal.