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3 and 4 .Determinants and Matrices
normal
Let $A$ be a non-zero periodic matrix with period $4$ and $A^{12} + B =I$, where $I$ is identity matrix and $B$ is any square matrix of same order as of $A$. Matrix product $AB$ is equal to
A
$I$
B
$A$
C
$A + I$
D
null matrix
Solution
$\mathrm{A}=\mathrm{A}^{4+1} \Rightarrow \mathrm{A}=\mathrm{A}^{5}=\mathrm{A}^{9}=\Lambda^{13}$
$\therefore \mathrm{A}^{12}+\mathrm{B}=\mathrm{I}$
$\mathrm{A}\left(\mathrm{A}^{12}+\mathrm{B}\right)=\mathrm{A} \Rightarrow \mathrm{A}^{13}+\mathrm{AB}=\mathrm{A} \ldots \Rightarrow \mathrm{AB}=0$
Standard 12
Mathematics
Similar Questions
Match the Statements / Expressions in Column $I$ with the Statements / Expressions in Column $II$ and indicate your answer by darkening the appropriate bubbles in the $4 \times 4$ matrix given in the $ORS$.
| Column $I$ | Column $II$ |
| $(A)$ The minimum value of $\frac{x^2+2 x+4}{x+2}$ is | $(p)$ $0$ |
| $(B)$ Let $A$ and $B$ be $3 \times 3$ matrices of real numbers, where $A$ is symmetric, $B$ is skewsymmetric, and $(A+B)(A-B)=(A-B)(A+B)$. If $(A B)^t=(-1)^k A B$, where $(A B)^t$ is the transpose of the matrix $A B$, then the possible values of $k$ are | $(q)$ $1$ |
| $(C)$ Let $\mathrm{a}=\log _3 \log _3 2$. An integer $\mathrm{k}$ satisfying $1<2^{\left(-k+3^{-2}\right)}<2$, must be less than | $(r)$ $2$ |
| $(D)$ If $\sin \theta=\cos \phi$, then the possible values of $\frac{1}{\pi}\left(\theta \pm \phi-\frac{\pi}{2}\right)$ are | $(s)$ $3$ |
