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Let $A$ and $B$ be two matrices of size $n \times n$ and with real-valued entries. Consider the following statements.

  1. If $AB = B$, then $A$ must be the identity matrix.
  2. If $A$ is an idempotent (i.e. $A^{2} = A$) nonsingular matrix, then $A$ must be the identity matrix.
  3. If $A^{-1} = A$, then $A$ must be the identity matrix.

Which of the above statements $\text{MUST}$ be true of $A$?

  1. $1, 2 $ and $3$
  2. Only $2$ and $3$
  3. Only $1$ and $2$
  4. Only $1$ and $3$
  5. Only $2$
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(1). If $AB =B$, then A must be the identity matrix. $\qquad \rightarrow$ False

If $B$ is invertible, then $B^{-1}$ exists, 

and $ABB^{-1} = BB^{-1} \implies AI = I \implies A=I$

But $B$ may not be invertible, in such case $A$ may or may not be identity matrix.

For example, If $A=B$, then we have $A^2 = A$, here $A$ can be a non-singular matrix.

For $A = B = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}$, we have $AB =B$, and $A$ is not identity.

Hence this is False.

 

(2). If $A$ is an idempotent nonsingular matrix, then $A$ must be the identity matrix. $\qquad \rightarrow$ True.

$A$ is nonsingular, means $A^{-1}$ exists,

$A^2 = A \implies AAA^{-1} = AA^{-1}  \implies AI = I \implies A=I$

Hence true.

 

(3).  If $A^{−1}=A$, then $A$ must be the identity matrix. $\qquad \rightarrow$ False.

$A$ is an Involutory matrix,  It not necessary that $A=I$

For example: $P = P^{-1} = \begin{bmatrix} 1 & 0 & 0\\ 0 & 0 &1 \\ 0&1&0 \end{bmatrix}$ $\qquad$ (Interchange row/column 2 and 3)

Hence False.

Option (E) is correct.

Answer:

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