List of top Mathematics Questions on Matrices asked in JEE Main

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

Let

$ A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \alpha & \beta \\ 0 & \beta & \alpha \end{bmatrix} $

and $|2A|^3 = 2^{21}$ where $\alpha, \beta \in \mathbb{Z}$. Then a value of $\alpha$ is:

Let \[ R = \begin{pmatrix} x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z \end{pmatrix} \text{ be a non-zero } 3 \times 3 \text{ matrix, where} \]

\[ x = \sin \theta, \quad y = \sin \left( \theta + \frac{2\pi}{3} \right), \quad z = \sin \left( \theta + \frac{4\pi}{3} \right) \]

and $ \theta \neq 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi $. For a square matrix $ M $, let $ \text{trace}(M) $ denote the sum of all the diagonal entries of $ M $. Then, among the statements:

$\text{Trace}(R) = 0$
If $\text{trace}(\text{adj}(\text{adj}(R))) = 0$, then $ R $ has exactly one non-zero entry.

Which of the following is true?

Let $ A = \begin{bmatrix} 2 & 0 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 1 \end{bmatrix} $, $ B = [B_1, B_2, B_3] $, where $ B_1, B_2, B_3 $ are column matrices, and $ AB_1 = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} $, $ AB_2 = \begin{bmatrix} 2 \\ 3 \\ 0 \end{bmatrix} $, $ AB_3 = \begin{bmatrix} 3 \\ 2 \\ 1 \end{bmatrix} $.
If $\alpha = |B|$ and $\beta$ is the sum of all the diagonal elements of B, then $\alpha^3 + \beta^3$ is equal to _____.

Let the system of equations \[x + 2y + 3z = 5, \quad 2x + 3y + z = 9, \quad 4x + 3y + \lambda z = \mu\]have an infinite number of solutions. Then $\lambda + 2\mu$ is equal to:

For $ \alpha, \beta \in \mathbb{R} $ and a natural number $ n $, let \[A_r = \begin{vmatrix} r & 1 & \frac{n^2}{2} + \alpha \\ 2r & 2 & n^2 - \beta \\3r - 2 & 3 & \frac{n(3n - 1)}{2} \end{vmatrix}.\]Then $ 2A_{10} - A_8 $ is:

Let A be a non-singular matrix of order 3. If \[ \text{det}\left(3 \text{adj}(2 \text{adj}((\text{det} A) A))\right) = 3^{-13} \cdot 2^{-10} \] and \[ \text{det}\left(3 \text{adj}(2 A)\right) = 2^m \cdot 3^n, \] then $ |3m + 2n| $ is equal to __________.

Let $ \lambda, \mu \in \mathbb{R} $. If the system of equations
$ 3x + 5y + \lambda z = 3 $
$ 7x + 11y - 9z = 2 $
$ 97x + 155y - 189z = \mu $
has infinitely many solutions, then $ \mu + 2\lambda $ is equal to:

Let $ A $ be a $ 3 \times 3 $ real matrix such that**
$A \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix} = 2 \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix}, \quad A \begin{pmatrix} 0 \\ 1 \\ 1 \end{pmatrix} = 4 \begin{pmatrix} -1 \\ 0 \\ 1 \end{pmatrix}, \quad A \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix} = 2 \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}.$
Then, the system $ (A - 3I) \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} $ has

If the system of linear equations
$ x - 2y + z = -4 $
$ 2x + \alpha y + 3z = 5 $
$ 3x - y + \beta z = 3 $
has infinitely many solutions, then $ 12\alpha + 13\beta $ is equal to

Consider the matrix $ f(x) = \begin{bmatrix} \cos x & -\sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1 \end{bmatrix} $.
Given below are two statements:
Statement I: $ f(-x) $ is the inverse of the matrix $ f(x) $.
Statement II: $ f(x) f(y) = f(x + y) $.
In the light of the above statements, choose the correct answer from the options given below:"

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

Let $A=\left[\begin{array}{cc}\frac{1}{\sqrt{10}} & \frac{3}{\sqrt{10}} \\ \frac{-3}{\sqrt{10}} & \frac{1}{\sqrt{10}}\end{array}\right]$ and $B=\left[\begin{array}{cc}1 & -i \\ 0 & 1\end{array}\right]$, where $i=\sqrt{-1}$ If $M = A ^{ T } B A$, then the inverse of the matrix $AM ^{2023} A ^{ T }$ is

If a point $P (\alpha, \beta, \gamma)$ satisfying $(\alpha\,\, \beta\,\, \gamma) \begin{pmatrix} 2 & 10 & 8 \\9 & 3 & 8 \\8 & 4 & 8\end{pmatrix}=(0\,\,0\,\,0) $ lies on the plane $2 x+4 y+3 z=5$, then $6 \alpha+9 \beta+7 \gamma$ is equal to :

Let the system of linear equations $ x+y+ kz =2$ $ 2 x+3 y-z=1 $ $ 3 x+4 y +2 z= k$ have infinitely many solutions Then the system $ ( k +1) x+(2 k -1) y=7$ $ (2 k +1) x+( k +5) y=10$ has: