JEE MAIN Matrices Previous Year Questions (PYQs) – Page 11 of 15

Let the system of linear equations4x + $\lambda$y + 2z = 0 ,2x $-$ y + z = 0 , $\mu$x + 2y + 3z = 0, $\lambda$, $\mu$$\in$R. has a non-trivial solution. Then which of the following is true?

If the following system of linear equations 2x + y + z = 5, x $-$ y + z = 3, x + y + az = b has no solution, then :

Consider the following system of equations \[ \begin{cases} \alpha x+2y+z=1,\\ 2\alpha x+3y+z=1,\\ 3x+\alpha y+2z=\beta \end{cases} \] for some $\alpha,\beta\in\mathbb{R}$. Then which of the following is NOT correct?

Let $A$ be a $3\times3$ real matrix such that \[ A\!\begin{pmatrix}1\\0\\1\end{pmatrix} =2\!\begin{pmatrix}1\\0\\1\end{pmatrix},\qquad A\!\begin{pmatrix}-1\\0\\1\end{pmatrix} =4\!\begin{pmatrix}-1\\0\\1\end{pmatrix},\qquad A\!\begin{pmatrix}0\\1\\0\end{pmatrix} =2\!\begin{pmatrix}0\\1\\0\end{pmatrix}. \] Then, the system $(A-3I)\!\begin{pmatrix}x\\y\\z\end{pmatrix}=\begin{pmatrix}1\\2\\3\end{pmatrix}$ has:

If ${a_r} = \cos {{2r\pi } \over 9} + i\sin {{2r\pi } \over 9}$, r = 1, 2, 3, ....., i = $\sqrt { - 1} $, then the determinant $\left| {\matrix{ {{a_1}} & {{a_2}} & {{a_3}} \cr {{a_4}} & {{a_5}} & {{a_6}} \cr {{a_7}} & {{a_8}} & {{a_9}} \cr } } \right|$ is equal to :<

$A = \begin{bmatrix} 5a & -b \\ 3 & 2 \end{bmatrix}$ and $A \,\text{adj}\, A = A\,A^{T}$, then $5a + b$ is equal to:

If $\alpha$ + $\beta$ + $\gamma$ = 2$\pi$, then the system of equations :- x + (cos $\gamma$)y + (cos $beta$)z = 0,(cos $\gamma$)x + y + (cos $\alpha$)z = 0(cos $\beta$)x + (cos $\alpha$)y + z = 0 has :

If the system of linear equations

2x + y $-$ z = 7

x $-$ 3y + 2z = 1

x + 4y + $\delta$z = k, where $\delta$, k $\in$ R has infinitely many solutions, then $\delta$ + k is equal to:

Let $A$ be a $3 \times 3$ real matrix such that $A^2(A-2 I)-4(A-I)=O$, where $I$ and $O$ are the identity and null matrices, respectively. If $A^5=\alpha A^2+\beta A+\gamma I$, where $\alpha, \beta$, and $\gamma$ are real constants, then $\alpha+\beta+\gamma$ is equal to :