(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.