JEE MAIN Matrices Previous Year Questions (PYQs) – Page 7 of 15

If the system of equations $x+y+z=6$ $2x+5y+\alpha z=\beta$ $x+2y+3z=14$ has infinitely many solutions, then $\alpha+\beta$ is equal to:

Let $A = \left[ {\matrix{ i & { - i} \cr { - i} & i \cr } } \right],i = \sqrt { - 1} $. Then, the system of linear equations ${A^8}\left[ {\matrix{ x \cr y \cr } } \right] = \left[ {\matrix{ 8 \cr {64} \cr } } \right]$ has :

If the system of equations

$\alpha$x + y + z = 5, x + 2y + 3z = 4, x + 3y + 5z = $\beta$

has infinitely many solutions, then the ordered pair ($\alpha$, $\beta$) is equal to :

Let $\mathrm{A}=\left[\begin{array}{cc}\frac{1}{\sqrt{2}} & -2 \\ 0 & 1\end{array}\right]$ and $\mathrm{P}=\left[\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta\end{array}\right], \theta>0$. If $\mathrm{B}=\mathrm{PAP}{ }^{\top}, \mathrm{C}=\mathrm{P}^{\top} \mathrm{B}^{10} \mathrm{P}$ and the sum of the diagonal elements of $C$ is $\frac{m}{n}$, where $\operatorname{gcd}(m, n)=1$, then $m+n$ is :

If $A = \begin{bmatrix} 2 & -3 \\ -4 & 1 \end{bmatrix}$, then $\operatorname{adj}(3A^{2} + 12A)$ is equal to :

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

Let $\mathbf{A}$ be a $2 \times 2$ matrix with real entries such that $\mathbf{A}' = \alpha \mathbf{A} + \mathbf{I}$, where $\alpha \in \mathbb{R} - \{-1, 1\}$. If $\det(\mathbf{A}^2 - \mathbf{A}) = 4$, then the sum of all possible values of $\alpha$ is equal to:

Let $A=\left[\begin{array}{lll}2 & a & 0 \\ 1 & 3 & 1 \\ 0 & 5 & b\end{array}\right]$. If $A^3=4 A^2-A-21 I$, where $I$ is the identity matrix of order $3 \times 3$, then $2 a+3 b$ is equal to

Let $A = \left( {\matrix{ {\cos \alpha } & { - \sin \alpha } \cr {\sin \alpha } & {\cos \alpha } \cr } } \right)$, ($\alpha $ $ \in $ R)
such that ${A^{32}} = \left( {\matrix{ 0 & { - 1} \cr 1 & 0 \cr } } \right)$ then a value of $\alpha $

Let a, b, c $ \in $ R be all non-zero and satisfy a³ + b³ + c³ = 2. If the matrix A = $\left( {\matrix{ a & b & c \cr b & c & a \cr c & a & b \cr } } \right)$ satisfies A^TA = I, then a value of abc can be :