Which of the following statements is incorrect for a square matrix $A$ where $|A| \neq 0$?

  • A
    If $A$ is a diagonal matrix,$A^{-1}$ will also be a diagonal matrix.
  • B
    If $A$ is a symmetric matrix,$A^{-1}$ will also be a symmetric matrix.
  • C
    If $A^{-1} = A$,then $A$ is an idempotent matrix.
  • D
    If $A^{-1} = A$,then $A$ is an involutory matrix.

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If $\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 3 \\ 0 & 1 \end{bmatrix} \dots \begin{bmatrix} 1 & n-1 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 78 \\ 0 & 1 \end{bmatrix}$,then the inverse of $\begin{bmatrix} 1 & n \\ 0 & 1 \end{bmatrix}$ is

If the inverse of the matrix $A = \begin{bmatrix} -1 & -3 & -2 \\ 0 & 1 & 2 \\ 3 & 4 & 5 \end{bmatrix}$ is $A^{-1} = \begin{bmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{bmatrix}$,then find the value of $a_1 + c_2 + b_3$.

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$A=\left[\begin{array}{rr}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta\end{array}\right]$ and $AB=BA=I$,then $B$ is equal to

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