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Question: If A and B are two square matrices such that AB = A and BA = B, then \({{A}^{2}}+{{B}^{2}}=\) a)....

If A and B are two square matrices such that AB = A and BA = B, then A2+B2={{A}^{2}}+{{B}^{2}}=
a). A + B
b). A – B
C). AB
d). 0

Explanation

Solution

We will use the property of matrix multiplication to solve the above question. We will use the associative property of matrix multiplication (i.e. A(BC) = (AB)C) and also AB is not equal to BA.

Complete step-by-step solution:
We will solve the above question by using the matrix multiplication properties. We will use the associative property of matrix multiplication.
Associative property of matrix multiplication states that A(BC) = (AB)C and we are also required to note that when we multiply any matrix by another matrix then the number of columns of the first matrix must be equal to the number of rows of the second matrix otherwise, we can’t multiply matrices.
We know from the question that AB = A and both A and B are square matrices. So, after multiplying both sides of the equation by A we will get:
(AB)A = AA
⇒A(BA)=A2\Rightarrow A\left( BA \right)={{A}^{2}}
And, from the question, we know that BA = B.
So, AB=A2AB={{A}^{2}} .
Also, AB = A. So, A=A2.............(1)A={{A}^{2}}.............(1)
Similarly, we will find the value of B2{{B}^{2}}.
Since, BA = B. So, after multiplying both sides of the equation by B we will get:
(BA)B=BB\left( BA \right)B=BB
From the associative property of matrix multiplication, we know that A(BC) = (AB)C.
⇒B(AB)=B2\Rightarrow B\left( AB \right)={{B}^{2}}
From the question, we know that AB = A.
So, BA=B2BA={{B}^{2}}
Since BA = B so we will get B=B2................(2)B={{B}^{2}}................(2)
After adding (1) and (2) we will get:
A2+B2=A+B{{A}^{2}}+{{B}^{2}}=A+B
Hence, option (a) is our required answer.

Note: Students are required to note that when we multiply any matrix equation by any other matrix on both the side of the equation and if we are multiplying it on the front of the equation of one side then it is necessary to multiply the matrix on another side of the equation on the front itself and if we are multiplying it on the back of the equation of one side then it is necessary to multiply the matrix on another side of the equation on the back itself.