Maths. Section B1Matrices

If A and B are two matrices such that AB = Aand BA=B, then A^2+ B^2 i…

If A and B are two matrices such that $AB = A$and $BA=B$, then $A^2+ B^2$ is equal to

Answer & explanation

Correct answer: option 3

Proceeding as in Example 2, we have

We have,

$A^2 = AA$

$⇒A^2 (AB) A$ $[∵ AB=A]$

$⇒A^2 = A (BA)$

$⇒A^2 = AB$ $[∵ BA=B]$

$⇒A^2 = A$ $[∵ AB = A]$

and,

$B^2 = BB$

$⇒B^2 = (BA) B$ $[∵ BA =B]$

$⇒B^2=B (AB)$

$⇒B^2=BA$ $[∵ AB = A]$

$⇒B^2 = B$ $[∵ BA =B]$

Now, we have

$A^2 = A$ and $B^2 = B$

$∴A^2+ B^2=A+B$