(B) Given $ABA^{-1} = B^2$. We calculate powers of $B$: $B^2 = ABA^{-1}$ $B^4 = (ABA^{-1})(ABA^{-1}) = AB^2A^{-1} = A(ABA^{-1})A^{-1} = A^2BA^{-2}$ $B

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(B) Given $ABA^{-1} = B^2$. We calculate powers of $B$: $B^2 = ABA^{-1}$ $B^4 = (ABA^{-1})(ABA^{-1}) = AB^2A^{-1} = A(ABA^{-1})A^{-1} = A^2BA^{-2}$ $B^8 = (A^2BA^{-2})(A^2BA^{-2}) = A^2B^2A^{-2} = A^2(ABA^{-1})A^{-2} = A^3BA^{-3}$ Continuing this pattern,$B^{2^k} = A^kBA^{-k}$. For $k=5$,$B^{2^5} = A^5BA^{-5}$. Since $A^5 = I$,we have $B^{32} = I B I^{-1} = B$. Thus,$B^{32} = B$,which implies $B^{31} = I$. The power of matrix $B$ is $31$,which lies between $28$ and $32$.

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

Let the minimum $m$ $(m \in Z^+)$ be defined as the power of a square matrix $A$ such that $A^m = I$. If $A^5 = I$ and $ABA^{-1} = B^2$,then the power of matrix $B$ such that $B^k = I$ is between:

A
$20$ and $24$
B
$28$ and $32$
C
$36$ and $40$
D
$44$ and $48$

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3 and 4 .Determinants and Matrices

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Mix Examples-Determinants and Matrices

$\det \left[ \begin{array}{ccc} \frac{a^2+b^2}{c} & c & c \\ a & \frac{b^2+c^2}{a} & a \\ b & b & \frac{c^2+a^2}{b} \end{array} \right] = $

MediumTS EAMCET 2024

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Let $A$ be a $3 \times 3$ matrix such that $A^2 - 5A + 7I = 0$.
Statement-$I$: ${A^{-1}} = \frac{1}{7}(5I - A)$.
Statement-$II$: The polynomial $A^3 - 2A^2 - 3A + I$ can be reduced to $5(A - 4I)$.

DifficultJEE MAIN 2016

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If $\begin{bmatrix} x & 4 & -1 \end{bmatrix} \begin{bmatrix} 2 & 1 & 0 \\ 1 & 0 & 2 \\ 0 & 2 & 4 \end{bmatrix} \begin{bmatrix} x \\ 4 \\ -1 \end{bmatrix} = 0$,then $x=$

EasyAP EAMCET 2023

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Let $ABC = I$. Then $tr(ABC + BCA + CAB)$ is (where the order of matrices $A, B, C$ is $3 \times 3$ and $tr(A)$ is the sum of the diagonal elements in $A$).

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Let $A = [a_{ij}]$,where $a_{ij} \in \mathbb{Z} \cap [0, 4]$ and $1 \leq i, j \leq 2$. The number of matrices $A$ such that the sum of all entries is a prime number $p \in (2, 13)$ is $........$.

DifficultJEE MAIN 2023

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Let the minimum $m$ $(m \in Z^+)$ be defined as the power of a square matrix $A$ such that $A^m = I$. If $A^5 = I$ and $ABA^{-1} = B^2$,then the power of matrix $B$ such that $B^k = I$ is between:

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$\det \left[ \begin{array}{ccc} \frac{a^2+b^2}{c} & c & c \\ a & \frac{b^2+c^2}{a} & a \\ b & b & \frac{c^2+a^2}{b} \end{array} \right] = $

Let $A$ be a $3 \times 3$ matrix such that $A^2 - 5A + 7I = 0$.Statement-$I$: ${A^{-1}} = \frac{1}{7}(5I - A)$.Statement-$II$: The polynomial $A^3 - 2A^2 - 3A + I$ can be reduced to $5(A - 4I)$.

If $\begin{bmatrix} x & 4 & -1 \end{bmatrix} \begin{bmatrix} 2 & 1 & 0 \\ 1 & 0 & 2 \\ 0 & 2 & 4 \end{bmatrix} \begin{bmatrix} x \\ 4 \\ -1 \end{bmatrix} = 0$,then $x=$

Let $ABC = I$. Then $tr(ABC + BCA + CAB)$ is (where the order of matrices $A, B, C$ is $3 \times 3$ and $tr(A)$ is the sum of the diagonal elements in $A$).

Let $A = [a_{ij}]$,where $a_{ij} \in \mathbb{Z} \cap [0, 4]$ and $1 \leq i, j \leq 2$. The number of matrices $A$ such that the sum of all entries is a prime number $p \in (2, 13)$ is $........$.

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Let $A$ be a $3 \times 3$ matrix such that $A^2 - 5A + 7I = 0$.
Statement-$I$: ${A^{-1}} = \frac{1}{7}(5I - A)$.
Statement-$II$: The polynomial $A^3 - 2A^2 - 3A + I$ can be reduced to $5(A - 4I)$.