(A) Given that $Q^r = I$ for some integer $r > 1$.Multiplying both sides by $Q^{-1}$ on the right,we get $Q^r Q^{-1} = I Q^{-1}$.This simplifies to $Q

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(A) Given that $Q^r = I$ for some integer $r > 1$.Multiplying both sides by $Q^{-1}$ on the right,we get $Q^r Q^{-1} = I Q^{-1}$.This simplifies to $Q^{r-1} = Q^{-1}$.Now,substitute this into the expression $P^{-1}Q^{r-1}P - P^{-1}Q^{-1}P$.Since $Q^{r-1} = Q^{-1}$,the expression becomes $P^{-1}Q^{-1}P - P^{-1}Q^{-1}P$.This results in $O$,where $O$ is the null matrix.

Class 12 Mathematics 3 and 4 .Determinants and Matrices Mix Examples-Determinants and Matrices

If $P$ and $Q$ are two non-singular matrices of the same order such that $Q^r = I$,for some integer $r > 1$,then $P^{-1}Q^{r-1}P - P^{-1}Q^{-1}P$ is equal to (where $I$ is the identity matrix and $O$ is the null matrix).

A
$O$
B
$2I$
C
$I$
D
$-I$

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3 and 4 .Determinants and Matrices

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Mix Examples-Determinants and Matrices

If $A$ is a skew-symmetric matrix of order $3$ and $X$ is another matrix of the same order,then $|XA + AX^T|$ is (where $|P|$ denotes the determinant of matrix $P$).

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Let $A$ and $B$ be orthogonal matrices and $\operatorname{det}(A) + \operatorname{det}(B) = 0$. Then

DifficultWBJEE 2023

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Let $A = \begin{bmatrix} 2 & 1 & 2 \\ 6 & 2 & 11 \\ 3 & 3 & 2 \end{bmatrix}$ and $P = \begin{bmatrix} 1 & 2 & 0 \\ 5 & 0 & 2 \\ 7 & 1 & 5 \end{bmatrix}$. The sum of the prime factors of $|P^{-1}AP - 2I|$ is equal to

DifficultJEE MAIN 2024

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Let $A = \begin{bmatrix} 0 & -2 \\ 2 & 0 \end{bmatrix}$. If $M$ and $N$ are two matrices given by $M = \sum_{k=1}^{10} A^{2k}$ and $N = \sum_{k=1}^{10} A^{2k-1}$,then $MN^2$ is

MediumJEE MAIN 2022

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Let $A$ and $B$ be two non-singular matrices of order $3$ such that $A + B = I$ and $A^{-1} + B^{-1} = 2I$. Then $|adj(4AB)|$ is equal to (where $adj(A)$ is the adjoint of matrix $A$):

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If $P$ and $Q$ are two non-singular matrices of the same order such that $Q^r = I$,for some integer $r > 1$,then $P^{-1}Q^{r-1}P - P^{-1}Q^{-1}P$ is equal to (where $I$ is the identity matrix and $O$ is the null matrix).

Similar Questions

If $A$ is a skew-symmetric matrix of order $3$ and $X$ is another matrix of the same order,then $|XA + AX^T|$ is (where $|P|$ denotes the determinant of matrix $P$).

Let $A$ and $B$ be orthogonal matrices and $\operatorname{det}(A) + \operatorname{det}(B) = 0$. Then

Let $A = \begin{bmatrix} 2 & 1 & 2 \\ 6 & 2 & 11 \\ 3 & 3 & 2 \end{bmatrix}$ and $P = \begin{bmatrix} 1 & 2 & 0 \\ 5 & 0 & 2 \\ 7 & 1 & 5 \end{bmatrix}$. The sum of the prime factors of $|P^{-1}AP - 2I|$ is equal to

Let $A = \begin{bmatrix} 0 & -2 \\ 2 & 0 \end{bmatrix}$. If $M$ and $N$ are two matrices given by $M = \sum_{k=1}^{10} A^{2k}$ and $N = \sum_{k=1}^{10} A^{2k-1}$,then $MN^2$ is

Let $A$ and $B$ be two non-singular matrices of order $3$ such that $A + B = I$ and $A^{-1} + B^{-1} = 2I$. Then $|adj(4AB)|$ is equal to (where $adj(A)$ is the adjoint of matrix $A$):

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