Question

If $A=\left[\begin{array}{ll}\frac{1}{\sqrt{5}}& \frac{2}{\sqrt{5}}\\ \frac{-2}{\sqrt{5}}& \frac{1}{\sqrt{5}}\end{array}\right],B=\left[\begin{array}{ll}1& 0\\ i& 1\end{array}\right],i=\sqrt{-1}$, and $Q=A^{T}BA$, then the inverse of the matrix $A{Q}^{2021}{A}^{\mathrm{T}}$ is equal to:

JEE Main 2021 (26 Aug Shift 1)

Options

• A: $\left[\begin{array}{cc}1& -2021\\ 0& 1\end{array}\right]$
• B: $\left[\begin{array}{cc}1& 0\\ -2021i& 1\end{array}\right]$
• C: $\left[\begin{array}{cc}\frac{1}{\sqrt{5}}& -2021\\ 2021& \frac{1}{\sqrt{5}}\end{array}\right]$
• D: $\left[\begin{array}{ccc}1& 0& \\ 2021& i& 1\end{array}\right]$

Question

Let $A=\left\{a_{ij}\right\}$ be a $3\times 3$ matrix, where $a_{ij}=\left\{\begin{array}{c}{\left(-1\right)}^{j-i} \mathrm{if} i<j\\ 2 \mathrm{if} i=j\\ {\left(-1\right)}^{i+j} \mathrm{if} i>j\end{array}\right.$ then $\det \left(3Adj\left(2A^{-1}\right)\right)$ is equal to ________.

JEE Main 2021 (20 Jul Shift 2)

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Question

Let $P=\left[\begin{array}{rrr}3& -1& -2\\ 2& 0& \alpha \\ 3& -5& 0\end{array}\right],$ where $\alpha \in R.$ Suppose $Q=\left[q_{ij}\right]$ is a matrix satisfying $PQ=k{I}_3$ for
some non-zero $k\in R.$ If $q_{23}=-\frac{k}{8}$ and $\left|Q\right|=\frac{k^2}{2}$, then ${\alpha }^2+k^2$ is equal to_________.

JEE Main 2021 (24 Feb Shift 1)

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Question

Let the matrix $A=\left[\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right]$ and the matrix $B_0=A^{49}+2A^{98}$. If $B_{\mathit{n}}=\mathrm{Adj}\left(B_{n-1}\right)$ for all $n\ge 1$, then $det\left(B_4\right)$ is equal to

JEE Main 2022 (28 Jul Shift 1)

Options

• A: $3^{28}$
• B: $3^{30}$
• C: $3^{32}$
• D: $3^{36}$

Question

The probability that a randomly chosen $2\times 2$ matrix with all the entries from the set of first $10$ primes, is singular, is equal to

JEE Main 2022 (29 Jun Shift 1)

Options

• A: $\frac{133}{{10}^4}$
• B: $\frac{19}{{10}^3}$
• C: $\frac{18}{{10}^3}$
• D: $\frac{271}{{10}^4}$

Question

Let $A=\left(\begin{array}{cc}2& -1\\ 0& 2\end{array}\right)$. If $B=I-{}^{5}C_1\left(adj A\right)+{}^{5}C_2(adj A{)}^2\mathit{-}\mathit{ }\mathit{.}\mathit{.}\mathit{.}\mathit{-}\mathit{ }{}^{\mathit{5}}C_{\mathit{5}}{(\mathrm{adj} A)}^{\mathit{5}}$, then the sum of all elements of the matrix $B$ is:

JEE Main 2022 (29 Jun Shift 2)

Options

• A: $-5$
• B: $-6$
• C: $-7$
• D: $-8$

Question

Let $A=\left[\begin{array}{ccc}2& -1& -1\\ 1& 0& -1\\ 1& -1& 0\end{array}\right]$ and $B=A-I$. If $\omega =\frac{\sqrt{3}\mathrm{i}-1}{2}$, then the number of elements in the set $\left\{n\in \left\{1,2,\dots ,100\right\}:A^n+{\left(\omega B\right)}^n=A+B\right\}$ is equal to _____ .

JEE Main 2022 (25 Jul Shift 1)

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Question

Let $A=\left[\begin{array}{lll}1& a& a\\ 0& 1& b\\ 0& 0& 1\end{array}\right],a,b\in \mathrm{ℝ}$. If for some $n\in N,A^n=\left[\begin{array}{ccc}1& 48& 2160\\ 0& 1& 96\\ 0& 0& 1\end{array}\right]$ then $n+a+b$ is equal to _______.

JEE Main 2022 (25 Jul Shift 2)

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Question

The number of matrices $A=\left[\begin{array}{ll}a& b\\ c& d\end{array}\right]$, where $a,b,c,\mathrm{d}\in \left\{-1,0,1,2,3,\dots \dots ,10\right\}$, such that $A=A^{-1}$, is ______.

JEE Main 2022 (26 Jul Shift 2)

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Question

Let $S$ be the set containing all $3\times 3$ matrices with entries from $\left\{-1,0,1\right\}$. The total number of matrices $A\in S$ such that the sum of all the diagonal elements of $A^{T}A$ is $6$ is ______.

JEE Main 2022 (27 Jul Shift 1)

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Question

Let $x=\left[\begin{array}{l}1\\ 1\\ 1\end{array}\right]$ and $A=\left[\begin{array}{ccc}-1& 2& 3\\ 0& 1& 6\\ 0& 0& -1\end{array}\right]$. For $k\in \mathrm{\mathbb{N}}$, if $X^{\text{'}}{A}^{k}X=33$, then $k$ is equal to

JEE Main 2022 (29 Jul Shift 2)

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Question

Let $A$ be a $3\times 3$ matrix having entries from the set $\left\{-1,0,1\right\}$. The number of all such matrices $A$ having sum of all the entries equal to $5$, is _____

JEE Main 2022 (25 Jun Shift 1)

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Question

Let $A,B,C$ be $3\times 3$ matrices such that $A$ is symmetric and $B$ and $C$ are skew-symmetric.

Consider the statements

$\left(S1\right) A^{13} B^{26}-B^{26} A^{13}$ is symmetric

$\left(\mathrm{S}2\right) A^{26}{C}^{13}-C^{13} A^{26}$ is symmetric

Then,

JEE Main 2023 (25 Jan Shift 2)

Options

• A: Only $S2$ is true
• B: Only $\mathrm{S}1$ is true
• C: Both $S1$ and $S2$ are false
• D: Both $S1$ and $S2$ are true

Question

The set of all values of $t\in \mathrm{ℝ}$, for which the matrix $\left[\begin{array}{ccc}{\mathrm{e}}^t& {\mathrm{e}}^{-t}\left(\sin t-2\cos t\right)& {\mathrm{e}}^{-t}\left(-2\sin t-\cos t\right)\\ {\mathrm{e}}^t& {\mathrm{e}}^{-t}\left(2\sin t+\cos t\right)& {\mathrm{e}}^{-t}\left(\sin t-2\cos t\right)\\ {\mathrm{e}}^t& {\mathrm{e}}^{-t}\cos t& {\mathrm{e}}^{-t}\sin t\end{array}\right]$ is invertible, is

JEE Main 2023 (29 Jan Shift 2)

Options

• A: $\left\{\left(2k+1\right)\frac{\pi }{2},k\in \mathbb{Z}\right\}$
• B: $\left\{k\pi +\frac{\pi }{4},k\in \mathbb{Z}\right\}$
• C: $\left\{k\pi ,k\in \mathbb{Z}\right\}$
• D: $\mathrm{ℝ}$

Question

Let $A=\left[\begin{array}{cc}m& n\\ p& q\end{array}\right], d=\left|A\right|\ne 0$ and $\left|A-\mathrm{d}\left(AdjA\right)\right|=0$. Then

JEE Main 2023 (30 Jan Shift 1)

Options

• A: ${\left(1+d\right)}^2={\left(m+q\right)}^2$
• B: $1+d^2={\left(m+q\right)}^2$
• C: ${\left(1+d\right)}^2=m^2+q^2$
• D: $1+d^2=m^2+q^2$

Question

Let $P$ be a square matrix such that $P^2=I-P$. For $\alpha , \beta , \gamma , \delta \in \mathrm{\mathbb{N}}$, if $P^{\alpha }+P^{\beta }=\gamma l-29P$ and $P^{\alpha }-P^{\beta }=\delta l-13P$, then $\alpha +\beta +\gamma -\delta$ is equal to

JEE Main 2023 (06 Apr Shift 2)

Options

• A: $18$
• B: $40$
• C: $22$
• D: $24$

Question

Let $P=\left[\begin{array}{cc}\frac{\sqrt{3}}{2}& \frac{1}{2}\\ -\frac{1}{2}& \frac{\sqrt{3}}{2}\end{array}\right], A=\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]$ and $Q=PA{P}^T$. If $P^T{Q}^{2007} P=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$then $2a+b-3c-4d$ is equal to

JEE Main 2023 (08 Apr Shift 1)

Options

• A: $2004$
• B: $2005$
• C: $2007$
• D: $2006$

Question

Let $A=\left[\begin{array}{cc}1& \frac{1}{51}\\ 0& 1\end{array}\right]$. If $B=\left[\begin{array}{cc}1& 2\\ -1& -1\end{array}\right] A\left[\begin{array}{cc}-1& -2\\ 1& 1\end{array}\right]$, then the sum of all the elements of the matrix $\sum _{n=1}^{50}{B}^n$ is equal to

JEE Main 2023 (12 Apr Shift 1)

Options

• A: $75$
• B: $125$
• C: $50$
• D: $100$

Question

Let $\alpha \in(0, \infty)$ and $A=\left[\begin{array}{lll}1 & 2 & \alpha \\ 1 & 0 & 1 \\ 0 & 1 & 2\end{array}\right]$. If $\operatorname{det}\left(\operatorname{adj}\left(2 A-A^T\right) \cdot \operatorname{adj}\left(A-2 A^T\right)\right)=2^8$, then $(\operatorname{det}(A))^2$ is equal to:

JEE Main 2024 (04 Apr Shift 1)

Options

• A: 36
• B: 16
• C: 1
• D: 49

Question

Let $A=\left[\begin{array}{ll}1 & 2 \\ 0 & 1\end{array}\right]$ and $B=I+\operatorname{adj}(A)+(\operatorname{adj} A)^2+\ldots+(\operatorname{adj} A)^{10}$. Then, the sum of all the elements of the matrix $B$ is:

JEE Main 2024 (04 Apr Shift 2)

Options

• A: -124
• B: 22
• C: -88
• D: -110

Question

Let $A$ and $B$ be two square matrices of order 3 such that $|A|=3$ and $|B|=2$. Then $\left|\mathrm{A}^{\mathrm{T}} \mathrm{A}(\operatorname{adj}(2 \mathrm{~A}))^{-1}(\operatorname{adj}(4 \mathrm{~B}))(\operatorname{adj}(\mathrm{AB}))^{-1} \mathrm{AA}^{\mathrm{T}}\right|$ is equal to :

JEE Main 2024 (05 Apr Shift 1)

Options

• A: 108
• B: 32
• C: 81
• D: 64

Question

If $A$ is a square matrix of order 3 such that $\operatorname{det}(A)=3$ and $\operatorname{det}\left(\operatorname{adj}\left(-4 \operatorname{adj}\left(-3 \operatorname{adj}\left(3 \operatorname{adj}\left((2 \mathrm{~A})^{-1}\right)\right)\right)\right)\right)=2^{\mathrm{m}} 3^{\mathrm{n}}$, then $\mathrm{m}+2 \mathrm{n}$ is equal to :

JEE Main 2024 (06 Apr Shift 2)

Options

• A: 2
• B: 3
• C: 6
• D: 4

Question

Let $B=\left[\begin{array}{ll}1 & 3 \\ 1 & 5\end{array}\right]$ and $A$ be a $2 \times 2$ matrix such that $A B^{-1}=A^{-1}$. If $B C B^{-1}=A$ and $C^4+\alpha C^2+\beta I=O$, then $2 \beta-\alpha$ is equal to

JEE Main 2024 (09 Apr Shift 2)

Options

• A: 16
• B: 2
• C: 8
• D: 10

Question

Let $\mathrm{A}$ be a $2\times 2$ real matrix and $\mathrm{I}$ be the identity matrix of order $2.$ If the roots of the equation $|\mathrm{A}-\mathrm{xI}|=0$ be $-1$ and $3,$ then the sum of the diagonal elements of the matrix ${\mathrm{A}}^2$ is _____.

JEE Main 2024 (27 Jan Shift 2)

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Question

Let $A=I_2-2M{M}^T,$ where $M$ is real matrix of order $2\times 1$ such that the relation $M^{T}M=I_1$ holds. If $\lambda$ is a real number such that the relation $AX=\lambda X$ holds for some non-zero real matrix $X$ of order $2\times 1,$ then the sum of squares of all possible values of $\lambda$ is equal to:

JEE Main 2024 (01 Feb Shift 2)

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