(C) matrix $A$ is called an involutory matrix if $A^2 = I$,which implies $A = A^{-1}$.$A$ matrix $A$ is called an idempotent matrix if $A^2 = A$.Given

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If $A^{-1} = A$,then $A$ is an idempotent matrix.

(C) matrix $A$ is called an involutory matrix if $A^2 = I$,which implies $A = A^{-1}$.$A$ matrix $A$ is called an idempotent matrix if $A^2 = A$.Given the condition $A^{-1} = A$,multiplying both sides by $A$ gives $A \cdot A^{-1} = A \cdot A$,which simplifies to $I = A^2$.Since $A^2 = I$,the matrix $A$ is an involutory matrix,not an idempotent matrix.Therefore,the statement in option $C$ is incorrect.

Class 12 Mathematics 3 and 4 .Determinants and Matrices Adjoint and inverse of matrices

Medium

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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3 and 4 .Determinants and Matrices

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Adjoint and inverse of matrices

Let $A$ be a $3 \times 3$ matrix such that $\operatorname{adj} A = \begin{bmatrix} 2 & -1 & 1 \\ -1 & 0 & 2 \\ 1 & -2 & -1 \end{bmatrix}$ and $B = \operatorname{adj}(\operatorname{adj} A)$. If $|A| = \lambda$ and $|(B^{-1})^T| = \mu$,then the ordered pair $(|\lambda|, \mu)$ is equal to

DifficultJEE MAIN 2020

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Let $A = \begin{bmatrix} 1 & -2 & 1 \\ -2 & 3 & 1 \\ 1 & 1 & 5 \end{bmatrix}$. Verify that $[adj A]^{-1} = adj(A^{-1})$.

Medium

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If ${A^2} - A + I = 0$,then ${A^{-1}} = $

EasyAIEEE 2005

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The inverse of the matrix $\left[\begin{array}{ccc}7 & -3 & -3 \\ -1 & 1 & 0 \\ -1 & 0 & 1\end{array}\right]$ is

EasyAP EAMCET 2008

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Find the inverse of the matrix $A = \left[\begin{array}{rrr}2 & -3 & 3 \\ 2 & 2 & 3 \\ 3 & -2 & 2\end{array}\right]$,if it exists.

Medium

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

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