First slide

Operations of matrix

Question

.If $A B = A$ and $B A = B,$ where $A c . \cdot B$ are square
matrices, then

Moderate

A

$B^{2} = B and A^{2} = A$
B

$B^{2} \neq B and A^{2} = A$
C

$A^{2} \neq A, B^{2} = B$
D

$A^{2} \neq A, B^{2} \neq B$

Solution

We have,
$A^{2} = A A$
$\begin{array}{l} \Rightarrow A^{2} = (A B) [∵ A B = A] \\ \Rightarrow A^{2} = A (B A) \\ \Rightarrow A^{2} = A B [∵ B A = B] \\ \Rightarrow A^{2} = A [∵ A B = A] \end{array}$
and,
$B^{2} = B B$
$\begin{array}{l} B^{2} = B B \\ \Rightarrow B^{2} = (B A) B [∵ B A = B] \\ \Rightarrow B^{2} = B (A B) \\ \Rightarrow B^{2} = B A [∵ A B = A] \\ \Rightarrow B^{2} = B [∵ A B = A] \end{array}$

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