Matrices Question 1
Question 1 - 24 January - Shift 1
If $A$ and $B$ are two non-zero $n \times n$ matrices such that $A^{2}+B=A^{2} B$, then
(1) $AB=I$
(2) $A^{2} B=I$
(3) $A^{2}=I$ or $B=I$
(4) $A^{2} B=BA^{2}$
Show Answer
Answer: (4)
Solution:
Formula: Properties of Matrix Multiplication, Properties of Matrix Addition
$A^{2}+B=A^{2} B$
$(A^{2}-I)(B-I)=I$
$A^{2}+B=A^{2} B$
$A^{2}(B-I)=B$
$A^{2}=B(B-I)^{-1}$
$A^{2}=B(A^{2}-I)$
$A^{2}=BA^{2}-B$
$A^{2}+B=BA^{2}$
$A^{2} B=BA^{2}$
