Question

# If A and B are two matrices such that ${\text{AB = B}}$and${\text{BA = A}}$, then${{\text{A}}^2} + {{\text{B}}^2}$ is equal to${\text{a}}{\text{. 2AB }} \\ {\text{b}}{\text{. 2BA}} \\ {\text{c}}{\text{. A}} + {\text{B}} \\ {\text{d}}{\text{. AB}} \\$

Hint: - In this question use the associative property of the matrix which is $X\left( {YZ} \right) = \left( {XY} \right)Z$.

Given:
$AB = {\text{ }}B{\text{ }}..............\left( 1 \right),{\text{ }}BA = A...............\left( 2 \right)$
Now we have to find out the value of ${A^2} + {B^2}$
$\Rightarrow {A^2} + {B^2} = AA + BB$
Now from equation (1) and (2) substitute the values of matrix B and A in above equation we have,
$\Rightarrow {A^2} + {B^2} = A\left( {BA} \right) + B\left( {AB} \right)$
Now, from the associative property of matrix which is $X\left( {YZ} \right) = \left( {XY} \right)Z$ we have,
$\Rightarrow {A^2} + {B^2} = \left( {AB} \right)A + \left( {BA} \right)B$ (Associative property)
Now, again from equation (1) and (2) substitute the values of matrix AB and BA in above equation we have,
$\Rightarrow {A^2} + {B^2} = BA + AB$
Now, again from equation (1) and (2) substitute the values of matrix AB and BA in above equation we have,
$\Rightarrow {A^2} + {B^2} = A + B$
Hence, option (c) is the correct answer.

Note: -In these types of questions the key concept we have to remember is that always remember the properties of multiplication of matrix which is stated above then simplify the matrix according to given conditions then apply the associative property of matrix to get the required answer.