If A and B are two square matrices of the same order such that AB = B and BA = A, then A2 + B2 is always equal to ...............

Question

If A and B are two square matrices of the same order such that AB = B and BA = A, then A2 + B2 is always equal to ............... (A) 2AB (B) 2 BA (C) I (D) A+B

Tardigrade Admin · Accepted Answer

Given, A B=B, B A=A...(i) Then, A2+B2=A ⋅ A+B ⋅ B =A(B A)+B(A B) [from Eq.(i)] =(A B) A+(B A) B (by commutative law) =B A+A B [from Eq. (i)] =A+B [from Eq.(i)]

Q. If $A$ and $B$ are two square matrices of the same order such that $A B = B$ and $B A = A,$ then $A^{2} + B^{2}$ is always equal to ...............

2AB

2 BA

I

A+B

Given, $A B = B, B A = A$ ...(i)
Then, $A^{2} + B^{2} = A \cdot A + B \cdot B$
$= A (B A) + B (A B)$ [from Eq.(i)]
$= (A B) A + (B A) B$ (by commutative law)
$= B A + A B$ [from Eq. (i)]
$= A + B$ [from Eq.(i)]

Q. If A and B are two square matrices of the same order such that AB=B and BA=A, then A2+B2 is always equal to ...............

2AB

2 BA

I

A+B

Given, AB=B,BA=A...(i) Then, A2+B2=A⋅A+B⋅B =A(BA)+B(AB) [from Eq.(i)] =(AB)A+(BA)B (by commutative law) =BA+AB [from Eq. (i)] =A+B [from Eq.(i)]

Q. If $A$ and $B$ are two square matrices of the same order such that $A B = B$ and $B A = A,$ then $A^{2} + B^{2}$ is always equal to ...............

Given, $A B = B, B A = A$ ...(i)
Then, $A^{2} + B^{2} = A \cdot A + B \cdot B$
$= A (B A) + B (A B)$ [from Eq.(i)]
$= (A B) A + (B A) B$ (by commutative law)
$= B A + A B$ [from Eq. (i)]
$= A + B$ [from Eq.(i)]