If the inverse of $\left[\begin{array}{ccc}-x & 14 x & 7 x \\ 0 & 1 & 0 \\ x & -4 x & -2 x\end{array}\right]$ is $\left[\begin{array}{ccc}2 & 0 & 7 \\ 0 & 1 & 0 \\ 1 & -2 & 1\end{array}\right]$, then $\left|\begin{array}{ccc}x & x+1 & x+2 \\ x+1 & x+2 & x+3 \\ x+2 & x+3 & x+4\end{array}\right|=$

If the system of equations $2 x+3 y-3 z=3, x+2 y+0 z=1 2 x-y+z=\beta$ has infinitely many solutions, then $\frac{\alpha}{\beta}-\frac{\beta}{\alpha}=$

A value of $\theta$ lying between 0 and $\pi / 2$ and satisfying $\left|\begin{array}{ccc}1+\sin ^2 \theta & \cos ^2 \theta & 4 \sin 4 \theta \\ \sin ^2 \theta & 1+\cos ^2 \theta & 4 \sin 4 \theta \\ \sin ^2 \theta & \cos ^2 \theta & 1+4 \sin 4 \theta\end{array}\right|=0$ is

If the system of equations $2 x+p y+6 z=8$, $x+2 y+q z=5$ and $x+y+3 z=4$ has infinitely many solutions, then $p=$

If $x^a y^b=e^m, x^c y^d=e^n, \Delta_1=\left|\begin{array}{ll}m & b \\ n & d\end{array}\right|$, $\Delta_2=\left|\begin{array}{cc}a & m \\ c & n\end{array}\right|, \Delta_3=\left|\begin{array}{cc}a & b \\ c & d\end{array}\right|$, then the values of $x$ and $y$ are respectively ( $e$ is the base of natural logarithm)

If $B$ is the inverse of a third order matrix $A$ and det $B=k$, then $(\operatorname{adj}(\operatorname{adj} \mathrm{A}))^{-1}=$

If $A=\left[\begin{array}{lll}2 & 2 & 1 \\ 1 & 3 & 1 \\ 1 & 2 & 2\end{array}\right]$ and $\alpha, \beta, \gamma$ are the roots of the equation represented by $|A-x I|=0$, then $\alpha^2+\beta^2+\gamma^2=$

If the values of $x, y$ and $z$ which satisfy the equations $2 x-3 y+2 z+15=0,3 x+y-z+2=0$ and $x-3 y-3 z+8=0$ simultaneously are $\alpha, \beta$ and $\gamma$ respectively, then

If $a$ is the determinant of the adjoint of the matrix $\left[\begin{array}{lll}1 & 1 & 2 \\ 1 & 2 & 3 \\ 2 & 3 & 3\end{array}\right]$ and $b$ is the determinant of the inverse of the matrix $\left[\begin{array}{ccc}1 & 2 & 3 \\ 4 & -3 & -1 \\ 2 & 1 & -4\end{array}\right]$, then $\frac{b+1}{18 b}=$

Consider two systems of 3 linear equations in 3 unknowns $A X=B$ and $C X=D$. If $A X=B$ has unique solution $D$ and $C X=D$ has unique solution $B$, then the solution of $\left(A-C^{-1}\right) X=0$ is

$f(x)$ is an $n$th degree polynomial satisfying $f(x)=\frac{1}{2}\left|\begin{array}{cc}f(x) & f\left(\frac{1}{x}\right)-f(x) \\ 1 & f\left(\frac{1}{x}\right)\end{array}\right|$. If $f(2)=33$, then the value of $f(3)$ is

If $P=\left[\begin{array}{lll}1 & \alpha & 3 \\ 1 & 3 & 3 \\ 2 & 4 & 4\end{array}\right]$ is the adjoint of a matrix $A$ and det $A=4$, then the value of $\alpha$ is

If $\alpha$ is a real root of the equation $x^3+6 x^2+5 x-42=0$, then the determinant of the matrix

$\left[\begin{array}{lll}\alpha-1 & \alpha+1 & \alpha+2 \\ \alpha-2 & \alpha+3 & \alpha-3 \\ \alpha+4 & \alpha-4 & \alpha+5\end{array}\right]$ is

The rank of the matrix $\left[\begin{array}{cccc}2 & -3 & 4 & 0 \\ 5 & -4 & 2 & 1 \\ 1 & -3 & 5 & -4\end{array}\right]$ is

If $A=\left[\begin{array}{ccc}1 & 2 & x \\ 4 & -1 & 7 \\ 2 & 4 & -6\end{array}\right]$ and the rank of $A$ is 2 , then the value of $x$ is equal to

$ \left|\begin{array}{ll} 2 & 1 \\ 3 & 1 \end{array}\right|+\left|\begin{array}{cc} 1 & \frac{1}{3} \\ 3 & 1 \end{array}\right|+\left|\begin{array}{cc} \frac{1}{2} & \frac{1}{9} \\ 3 & 1 \end{array}\right|+\left|\begin{array}{cc} \frac{1}{4} & \frac{1}{27} \\ 3 & 1 \end{array}\right|+\ldots \infty= $

If $A=\left[\begin{array}{lll}1 & 2 & 3 \\ 1 & 3 & 5 \\ 2 & 1 & 6\end{array}\right]$ and $|\operatorname{adj}(\operatorname{adj} A)|(\operatorname{adj} A)^{-1}=k A$, then $k=$

If the values $x=\alpha, y=\beta, z=\gamma$ satisfy all the 3 equations $x+2 y+3 z=4,3 x+y+z=3$ and $x+3 y+3 z=2$, then $3 \alpha+\gamma=$

The number of solutions of the system of equations $2 x+y-z=7, x-3 y+2 z=1, x+4 y-3 z=5$ is

The value of $p$ and $q$ is that system of equations $2 x+p y+6 z=8, x+2 y+q z=5$ and $x+y+3 z=4$ may have no solution are

$A$ is the set of all matrices of order 3 with entries 0 or 1 only. $B$ is the subset of $A$ consisting of all matrices with determinant value 1 . If $C$ is the subset of $A$ consisting of all matrices with determinant value -1 , then

Consider the matrices $A=\left[\begin{array}{ccc}x & y & 0 \\ -3 & 1 & 2 \\ 1 & -2 & z\end{array}\right]$ and $B=\left[\begin{array}{ccc}1 & -2 & -2 \\ 2 & 0 & 1 \\ 2 & 1 & 0\end{array}\right]$

If the cofactors of the elements $z, 1$ in 3rd row and $x$ of $A$ are $9,4,3$, respectively then $A B=$

If $A=\left[\begin{array}{ccc}1 & 2 & -2 \\ 2 & -1 & 2 \\ -1 & 1 & -2\end{array}\right]$, then $A+2 A^{-1}=$

If $A=\left[\begin{array}{ccc}a & b & c \\ d & e & f \\ l & m & n\end{array}\right]$ is a matrix such that $|A|>0$ and $\operatorname{adj}(A)=\left[\begin{array}{ccc}0 & 4 & -6 \\ 10 & 8 & 0 \\ 2 & 4 & -4\end{array}\right]$, then $\frac{c d}{f b}+\frac{\ln }{e m}=$

In solving a system of linear equations $A X=B$ by Cramer's rule, in the usual notation, if $\Delta_1=\left|\begin{array}{ccc}-11 & 1 & -7 \\ -4 & 1 & -2 \\ 5 & 1 & 1\end{array}\right|$ and $\Delta_3=\left|\begin{array}{ccc}4 & 1 & -11 \\ 1 & 1 & -4 \\ 4 & 1 & 5\end{array}\right|$, then $X=$

If $A$ and $B$ are both $3 \times 3$ matrices, then which of the following statements are true?

(i) $A B=0 \Rightarrow A=0$ or $B=0$

(ii) $A B=I_3 \Rightarrow A^{-1}=B$

(iii) $(A-B)^2=A^2-2 A B+B^2$

$A=\left[\begin{array}{ccc}1 & -1 & 2 \\ -2 & 3 & -3\end{array}\right]$ is the given matrix and $A^T$ represents the transpose of $A$, then $A A^T-A-A^T=$

If $A=\left[\begin{array}{ccc}x & 2 & 1 \\ -2 & y & 0 \\ 2 & 0 & -1\end{array}\right], x$ and $y$ are non-zero numbers, trace of $A=0$ and determinant of $A=-6$, then the minor of the elements 1 of $A$ is

If the system of equations $a_1 x+b_1 y+c_1 z=0, a_2 x+b_2 y+c_2 z=0$ and $a_3 x+b_3 y+c_3 z=0$ has only trivial solution, then the rank of $\left[\begin{array}{lll}a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3\end{array}\right]$ is

If $\alpha, \beta$ and $\gamma(\alpha<\beta<\gamma)$ are the values of $x$ such that $\left[\begin{array}{ccc}x-2 & 0 & 1 \\ 1 & x+3 & 2 \\ 2 & 0 & 2 x-1\end{array}\right]$ is a singular matrix, then $2 \alpha+3 \beta+4 \gamma$ is equal to

If the set of equations $x+2 y+3 z=6, x+3 y+5 z=9$, $2 x+5 y+a z=b$ has unique solution, then

If $P$ and $Q$ are two $3 \times 3$ matrices such that $|P Q|=1$ and $|P|=9$, then the determinant of adjoint of the matrix $P$. $\operatorname{adj} 3 Q$ is

If $A=\left[\begin{array}{lll}a & 1 & 2 \\ 1 & 2 & b \\ c & 1 & 3\end{array}\right]$ and $\operatorname{adj} A=\left[\begin{array}{ccc}7 & -1 & -5 \\ -3 & 9 & 5 \\ 1 & -3 & 5\end{array}\right]$, then $a^2+b^2+c^2=$

Assertion (A) : If $B$ is a $3 \times 3$ matrix and $|B|=6$, then $|\operatorname{adj}(B)|=36$

Reason (R) : If $B$ is a square matrix of order $n$, then $|\operatorname{adj}(B)|=|B|^n$

While solving a system of linear equations $A X=B$ using Cramer's rule with the usual notation if

$ \Delta=\left|\begin{array}{ccc} 1 & 1 & 1 \\ 2 & -1 & 2 \\ -1 & 1 & 5 \end{array}\right|, \Delta_1=\left|\begin{array}{ccc} 5 & 1 & 1 \\ 4 & -1 & 2 \\ 11 & 1 & 5 \end{array}\right| \text { and } X=\left[\begin{array}{l} \alpha \\ 2 \\ \beta \end{array}\right] \text {, then } \alpha^2+\beta^2= $

If $A=\left[\begin{array}{lll}3 & -3 & 4 \\ 2 & -3 & 4 \\ 0 & -1 & 1\end{array}\right]$, then $A A^T$ is a

If $A X=D$ represents the system of simultaneous linear equations $x+y+z=6, 5 x-y+2 z=3$ and $2 x+y-z=-5$, then (Adj $A$) $D=$

If $A=\left[\begin{array}{ll}1 & 0 \\ 2 & 1\end{array}\right], B=\left[\begin{array}{ll}1 & 3 \\ 0 & 1\end{array}\right]$, then $\operatorname{det}\left(A^6+B^6\right)=$

Let $G(x)=\left[\begin{array}{ccc}\cos x & -\sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1\end{array}\right]$. If $x+y=0$ then $G(x) G(y)=$

If $A=\left[\begin{array}{cc}2 & -3 \\ -4 & 1\end{array}\right]$, then $\left(A^T\right)^2+(12 A)^T=$

If $a, b, c$ are respectively the 5 th, 8 th, 13 th terms of an arithmetic progression, then $\left|\begin{array}{ccc}a & 5 & 1 \\ b & 8 & 1 \\ c & 13 & 1\end{array}\right|=$

If $A=\left[\begin{array}{ccc}1 & 0 & 0 \\ a & -1 & 0 \\ b & c & 1\end{array}\right]$ is such that $A^2=I$, then

Let $A=\left[\begin{array}{ccc}-2 & x & 1 \\ x & 1 & 1 \\ 2 & 3 & -1\end{array}\right]$. If the roots of the equation $\operatorname{det} A=0$ are $l, m$ then $l^3-m^3=$

For $i=1,2,3$ and $j=1,23$ If $a_i^2+b_i^2+c_i^2=1, a_i a_j+b_i b_j+c_i c_j=0, \forall i \neq j$ and $A=\left[\begin{array}{lll}a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3\end{array}\right]$, then $\operatorname{det}\left(A A^T\right)=$

If $A=\frac{1}{7}\left[\begin{array}{ccc}3 & -2 & 6 \\ -6 & -3 & 2 \\ -2 & 6 & 3\end{array}\right]$, then

If $A=\left[\begin{array}{cc}\alpha^2 & 5 \\ 5 & -\alpha\end{array}\right]$ and $\operatorname{det}\left(A^{10}\right)=1024$, then $\alpha=$

Let $A=\left[\begin{array}{ccc}5 & \sin ^2 \theta & \cos ^2 \theta \\ -\sin ^2 \theta & -5 & 1 \\ \cos ^2 \theta & 1 & 5\end{array}\right]$. Then, maximum value of $\operatorname{det}(A)$ is

If $\frac{x^4+24 x^2+28}{\left(x^2+1\right)^3}=\frac{A x+B}{x^2+1}$ $+\frac{C x+D}{\left(x^2+1\right)^2}+\frac{E x+F}{\left(x^2+1\right)^3},$ then the value of $A+B+C+D+E+F=$

If $k \in R$ and $\operatorname{det} A=\left|\begin{array}{lll}a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3\end{array}\right|=k$, then $\operatorname{det} B=\left|\begin{array}{ccc}a_1 & b_1 & c_1 \\ a_2+2 a_1 & b_2+2 b_1 & c_2+2 c_1 \\ a_3 & b_3 & c_3\end{array}\right|$ is equal to

If $A=\left[\begin{array}{llll}\sqrt{2020} & \sqrt{2021} & \sqrt{2021} & \sqrt{2023} \\ \sqrt{4040} & \sqrt{4042} & \sqrt{4044} & \sqrt{4046} \\ \sqrt{6060} & \sqrt{6063} & \sqrt{6066} & \sqrt{6069} \\ \sqrt{8080} & \sqrt{8084} & \sqrt{8088} & \sqrt{8092}\end{array}\right]$, then the rank of $A$ is

If $\left|\begin{array}{lll}x & x^2 & 1+x^3 \\ y & y^2 & 1+y^3 \\ z & z^2 & 1+z^3\end{array}\right|=0$ and $x, y$ and $z$ are all distinct, then $x y z$ is equal to

Let A be a $n\times n$ matrix such that A is upper-triangular. Then, $adj (A)$ is equal to

If $f(x)=\left|\begin{array}{ccc}x & x^2 & x^3 \\ 1 & 2 x & 3 x^2 \\ 0 & 2 & 6 x\end{array}\right|$, then the ratio $f^{\prime \prime}(x): f^{\prime}(x)$ is equal to

The trace of the matrix $A=\left[\begin{array}{ccc}1 & -5 & 7 \\ 0 & 7 & 9 \\ 11 & 8 & 9\end{array}\right]$ is

If $A, B$ and $C$ are the angles of a triangle, then the system of equations $-x+y \cos C+z \cos B=0, x \cos C-y+z \cos A=0$ and $x \cos B+y \cos A-z=0$

If $\left[\begin{array}{cc}1 & -\tan \theta \\ \tan \theta & 1\end{array}\right]\left[\begin{array}{cc}1 & \tan \theta \\ -\tan \theta & 1\end{array}\right]^{-1} =\left[\begin{array}{cc}a & -b \\ b & a\end{array}\right]$, then

What is the value of $\left|\begin{array}{ccc}a & b & c \\ a-b & b-c & c-a \\ b+c & c+a & a+b\end{array}\right|$ ?

The value of $\left|\begin{array}{ccc}b+c & a & a \\ b & c+a & b \\ c & c & a+b\end{array}\right|$ is

Let $A, B, C, D$ be square real matrices such that $C^T=D A B, D^{\mathrm{T}}=A B C$ and $S=A B C D$, then $S^2$ is equal to

$A=\left[\begin{array}{ccc}a^2 & 15 & 31 \\ 12 & b^2 & 41 \\ 35 & 61 & c^2\end{array}\right]$ and $B=\left[\begin{array}{ccc}2 a & 3 & 5 \\ 2 & 2 b & 8 \\ 1 & 4 & 2 c-3\end{array}\right]$ are two matrices such that the sum of the principal diagonal elements of both $A$ and $B$ are equal, then the product of the principal diagonal elements of $B$ is

Let $a, b$ and $c$ be such that $b+c \neq 0$ and $\begin{aligned} & \left|\begin{array}{ccc} a & a+1 & a-1 \\ -b & b+1 & b-1 \\ c & c-1 & c+1 \end{array}\right| \\ & +\left|\begin{array}{ccc} a+1 & b+1 & c-1 \\ a-1 & b-1 & c+1 \\ (-1)^{n+2} a & (-1)^{n-1} b & (-1)^n c \end{array}\right|=0 \text {, } \\ & \end{aligned}$

then the value of $n$ is

The equation whose roots are the values of the equation $\left| {\matrix{ 1 & { - 3} & 1 \cr 1 & 6 & 4 \cr 1 & {3x} & {{x^2}} \cr } } \right| = 0$ is

Let a and b be non-zero real numbers such that $ab=5/2$ and given $A = \left[ {\matrix{ a & { - b} \cr b & a \cr } } \right]$ and $A{A^T} = 20I$ ($l$ is unit matrix), then the equation whose roots are a and b is

If $A=\left[\begin{array}{ccc}1 & -1 & 1 \\ 2 & 1 & -3 \\ 1 & 1 & 1\end{array}\right], 10 B=\left[\begin{array}{ccc}4 & 2 & 2 \\ -5 & 0 & \alpha \\ 1 & -2 & 3\end{array}\right]$ and $B=A^{-1}$, then the value of $\alpha$ is

The rank of the matrix $\left[\begin{array}{ccc}4 & 2 & (1-x) \\ 5 & k & 1 \\ 6 & 3 & (1+x)\end{array}\right]$ is 1 , then,

If $a_1, a_2, \ldots . a_9$ are in GP, then $\left|\begin{array}{lll}\log a_1 & \log a_2 & \log a_3 \\ \log a_4 & \log a_5 & \log a_6 \\ \log a_7 & \log a_8 & \log a_9\end{array}\right|$ is equal to

If $\mathbf{a}=2 \hat{\mathbf{i}}+\hat{\mathbf{j}}+3 \hat{\mathbf{k}}, \mathbf{b}=\hat{\mathbf{i}}+3 \hat{\mathbf{j}}-\hat{\mathbf{k}}$ and $\mathbf{c}=3 \hat{\mathbf{i}}-\hat{\mathbf{j}}-2 \hat{\mathbf{k}}$, then the value of $\left|\begin{array}{ccc}\mathbf{a} \cdot \mathbf{a} & \mathbf{a} \cdot \mathbf{b} & \mathbf{a} \cdot \mathbf{c} \\ \mathbf{b} \cdot \mathbf{a} & \mathbf{b} \cdot \mathbf{b} & \mathbf{b} \cdot \mathbf{c} \\ \mathbf{c} \cdot \mathbf{a} & \mathbf{c} \cdot \mathbf{b} & \mathbf{c} \cdot \mathbf{c}\end{array}\right|$ is equal to