JEE MAIN Matrices Previous Year Questions (PYQs) – Page 13 of 15

Consider the system of linear equations$-$x + y + 2z = 0 3x $-$ ay + 5z = 12x, $-$ 2y $-$ az = 7, Let S₁ be the set of all a$\in$R for which the system is inconsistent and S₂ be the set of all a$\in$R for which the system has infinitely many solutions. If n(S₁) and n(S₂) denote the number of elements in S₁ and S₂ respectively, then

If $A=\begin{bmatrix}\sqrt2&1\\-1&\sqrt2\end{bmatrix}$, $B=\begin{bmatrix}1&0\\1&1\end{bmatrix}$, $C=ABA^{\mathrm T}$ and $X=A^{\mathrm T}C^{2}A$, then $\det X$ is equal to:

Let ${J_{n,m}} = \int\limits_0^{{1 \over 2}} {{{{x^n}} \over {{x^m} - 1}}dx} $, $\forall$ n > m and n, m $\in$$ N. Consider a matrix $A = {[{a_{ij}}]_{3 \times 3}}$ where $${a_{ij}} = \left\{ {\matrix{ {{j_{6 + i,3}} - {j_{i + 3,3}},} & {i \le j} \cr {0,} & {i > j} \cr } } \right.$. Then $\left| {adj{A^{ - 1}}} \right|$ is :

If the system of equations
x+y+z=2
2x+4y–z=6
3x+2y+$\lambda $z=$\mu $
has infinitely many solutions, then

For $\alpha,\beta\in\mathbb{R}$, suppose the system of linear equations $\begin{aligned} x-y+z&=5,\\ 2x+2y+\alpha z&=8,\\ 3x-y+4z&=\beta \end{aligned}$ has infinitely many solutions. Then $\alpha$ and $\beta$ are the roots of:

Let y = y(x) satisfies the equation ${{dy} \over {dx}} - |A| = 0$, for all x > 0, where $A = \left[ {\matrix{ y & {\sin x} & 1 \cr 0 & { - 1} & 1 \cr 2 & 0 & {{1 \over x}} \cr } } \right]$. If $y(\pi ) = \pi + 2$, then the value of $y\left( {{\pi \over 2}} \right)$ is :

Suppose the vectors x₁, x₂ and x₃ are the solutions of the system of linear equations, Ax = b when the vector b on the right side is equal to b₁, b₂ and b₃ respectively. if${x_1} = \left[ {\matrix{ 1 \cr 1 \cr 1 \cr } } \right]$, ${x_2} = \left[ {\matrix{ 0 \cr 2 \cr 1 \cr } } \right]$, ${x_3} = \left[ {\matrix{ 0 \cr 0 \cr 1 \cr } } \right]$${b_1} = \left[ {\matrix{ 1 \cr 0 \cr 0 \cr } } \right]$, ${b_2} = \left[ {\matrix{ 0 \cr 2 \cr 0 \cr } } \right]$ and ${b_3} = \left[ {\matrix{ 0 \cr 0 \cr 2 \cr } } \right]$, then the determinant of A is equal to :

The value of k $\in$R, for which the following system of linear equations. 3x $-$ y + 4z = 3,x + 2y $-$ 3z = $-$2, 6x + 5y + kz = $-$3,has infinitely many solutions, is :

If $A = \begin{bmatrix} -4 & -1 \\ 3 & 1 \end{bmatrix}$, then the determinant of the matrix $(A^{2016} - 2A^{2015} - A^{2014})$ is: