Equality of matrices, Operations on Matrices/Algebra of Matrices, Practice Problems & FAQs

Just as we perform algebraic operations like addition, subtraction & multiplication on numbers, we can perform these operations on Matrices too. But now the question arises “ How to perform these operations? ” Let us now try to understand this in detail with the help of this article.

Table of contents:

• Equality of matrices
• Operations on Matrices/Algebra of Matrices
• Practice problems
• FAQs

Equality of matrices

Two matrices A and B are equal if the order of both the matrices are equal as well as their corresponding elements are equal.

Let A=aijmn and B=bijpq, then A=B if 

(i) m=p & n=q (ii) aij=bij i,j

Operations on Matrices/Algebra of Matrices

Addition, subtraction, and multiplication are the three basic operations which we can perform on matrices. But there are certain conditions that must be satisfied before we can perform these operations.

Addition of matrices

Two matrices A and B can be added, if the order of both the matrices is the same.The resultant matrix is obtained by adding the corresponding elements of both matrices.

If A=aijmn , B=bijmn then 

C=A+B , C=cijmn where cij=aij+bij i,j

Example:

Then, 

Properties of Matrix Addition:

The addition operation on matrices has the following properties:

• Commutative Property: If P and Q are two matrices of order m ✕ n, then the addition of the two matrices is commutative, i.e., P + Q= Q +P
• Associative Property: If P, Q and R are three matrices of order m ✕ n, then the addition of the three matrices is associative, i.e., P+ (Q+R) = (P+Q)+R
• Additive identity:If A is m ✕ n order matrix, then the additive identity of A will be zero matrix of same order, such that, A + O = A ( where O is an additive identity)
• Additive inverse: If A is any matrix of order m ✕ n, then the additive inverse of A will be (-A) matrix of the same order, such that, A + (-A) = O. Here, -A matrix is obtained by changing the sign of every element of A.

Subtraction of matrices

One matrix B and can be subtracted from an another matrix A ,if the order of both the matrices is the same. The resultant matrix is obtained by subtracting the corresponding elements of both matrices.

If A=aijmn , B=bijmn then 

C=A-B , C=cijmn where cij=aij-bij i,j

Example:

Then, 

Note: Subtraction of two matrices is not commutative i.e. A-BB-A

Multiplication of a matrix by a scalar

Let k be a scalar (real or complex) and A=aijmnbe a matrix ,then kA=bijmnwhere bij=kaij i and j .This means that every element of the matrix gets multiplied by the scalar number. Example:

. Then, 

Note:If A is scalar matrix,then A=I where is diagonal element and I is an identity matrix

Properties of Scalar multiplication:

If A & B are two comparable matrices having order m ✕ n then,

• kA=Ak ; k is a scalar
• k(AB)=kAkB
• (k1k2)A=k1Ak2A ; k1,k2 are scalars 
• k(A)=(k)A=(kA); k, are scalars

Practice Problems

Example: Find the value of x such that A=B

Solution:

Given A=B

Equating the corresponding elements, we obtain

cos x =-12 & sin x =12

Now, cos x<0 & sin x>0x lies in the Second Quadrant.

tan x= sin xcos x=-1

x=34 (Since x lies in second quadrant)

Example: Find the value of a, b, c and d, if

Solution:

Given   

If two matrices are equal then all their corresponding elements are also equal

a-b = -1 … (i)

2a + c = 5 … (ii)

2a-b=0 … (iii)

3c + d = 13 … (iv)

Subtracting equation (i) from (iii), we get

(2a-b)-(a-b)=0-(-1)

 a = 1

Substituting the value of a in (i), we get 1-b =-1 b=2

Substituting the value of a in (ii), we get 2+c= 5 c=3 

Substituting the value of c in (iv), we get 9+d=13 d=4

Hence a=1,b=2,c=3,d=4.

Example: Find the values of a and b if

Solution:

Equating the corresponding elements, we get

b+2=5 b=3 & 2a+7=8 a=12

Example: Find P and Q if

Solution:

Given,

Adding equations (i) & (ii), we get

Subtracting (ii) from (i), we get

 FAQs

Q 1. Can we divide two matrices of the same order?

Answer: The division of matrices is not defined.

Q 2. What is the application of algebra of matrices?

Answer: The most important application of algebra of matrices is in solving equations involving matrices.

Q 3. Is there a matrix, which is additive inverse to itself?

Answer: Yes, the Null matrix is the only matrix that is additive inverse to itself.

Q 4. Is the subtraction of matrices associative?

Answer: The subtraction of matrices is neither associative nor commutative.