List of top Mathematics Questions on Matrices asked in JEE Main

For the matrices $ A = \begin{bmatrix} 3 & -4 \\ 1 & -1 \end{bmatrix} $ and $ B = \begin{bmatrix} -29 & 49 \\ -13 & 18 \end{bmatrix} $, if $ (A^{15} + B) \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 0\\ 0 \end{bmatrix} $, then among the following which one is true?}

Let the relation $ R $ on the set $ M = \{1, 2, 3, \ldots, 16\} $ be given by \[ R = \{(x, y) : 4y = 5x - 3,\; x, y \in M\}. \] Then the minimum number of elements required to be added in $ R $, in order to make the relation symmetric, is equal to

The number of $3\times2$ matrices $A$, which can be formed using the elements of the set $\{-2,-1,0,1,2\}$ such that the sum of all the diagonal elements of $A^{T}A$ is $5$, is

Let matrix \[ A=\begin{pmatrix} 3 & -4\\ 1 & -1 \end{pmatrix} \] and $A^{100} = 100B + I$. Find the sum of all the elements in $B^{100}$.

If \[ X=\begin{bmatrix}x\\y\\z\end{bmatrix} \] is a solution of the system of equations $AX=B$, where \[ \text{adj }A= \begin{bmatrix} 4 & 2 & 2\\ -5 & 0 & 5\\ 1 & -2 & 3 \end{bmatrix} \quad \text{and} \quad B=\begin{bmatrix}4\\0\\2\end{bmatrix}, \] then $|x+y+z|$ is equal to

Let $A$ be a matrix of order $3 \times 3$ and $|A| = 5$. If \[ \left|\,2\,\text{adj}\big(3A\,\text{adj}(2A)\big)\right| = 2^{\alpha}\cdot 3^{\beta}\cdot 5^{\gamma}, \quad \alpha, \beta, \gamma \in \mathbb{N}, \] then the value of $\alpha + \beta + \gamma$ is:

Let \[ f(x)=\int \frac{7x^{10}+9x^8}{(1+x^2+2x^9)^2}\,dx \] and $f(1)=\frac14$. Given that

and $B=\operatorname{adj}(\operatorname{adj}A)$, if $|B|=81$, find the value of $\alpha^2$ (where $\alpha\in\mathbb{R}$).

If \[ A=\begin{bmatrix}2 & 3\\3 & 5\end{bmatrix}, \] then find the value of \[ \left|A^{2025}-3A^{2024}+A^{2023}\right|. \]

Find the number of matrices $A$ of order $3\times 2$ whose elements are from the set $\{\pm2,\pm1,0\}$, if $\mathrm{Tr}(A^TA)=5$.

Let \[ f(x)=\int \frac{7x^{10}+9x^8}{(1+x^2+2x^9)^2}\,dx \] and $f(1)=\frac14$. Given that

If A = $\begin{bmatrix} 2 & 3 \\ 3 & 5 \end{bmatrix}$ then value of det($A^{2025} - 3A^{2024} + A^{2023}$) :

Let $ A = \begin{bmatrix} 3 & -4 \\ 1 & -1 \end{bmatrix} $ and $ B = \begin{bmatrix} 29 & 49 \\1 & 2 \end{bmatrix} $. If $ (A^5 + B)\begin{bmatrix} x\\ y \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix} $, then find $ (x, y) $.

Let $ A $ be a matrix of order $ 3 \times 3 $ and $ |A| = 5 $. If $ |2 \, \text{adj}(3A \, \text{adj}(2A))| = 2^{\alpha} \cdot 3^{\beta} \cdot 5^{\gamma}, \quad \alpha, \beta, \gamma \in \mathbb{N} $ then $ \alpha + \beta + \gamma $ is equal to

Let $ A = \begin{bmatrix} \alpha & -1 \\ 6 & \beta \end{bmatrix} , \ \alpha > 0 $, such that $ \det(A) = 0 $ and $ \alpha + \beta = 1. $ If $ I $ denotes the $ 2 \times 2 $ identity matrix, then the matrix $ (I + A)^8 $ is:

Let $A = \begin{bmatrix} \cos \theta & 0 & -\sin \theta \\ 0 & 1 & 0 \\ \sin \theta & 0 & \cos \theta \end{bmatrix}$. If for some $\theta \in (0, \pi)$, $A^2 = A^T$, then the sum of the diagonal elements of the matrix $(A + I)^3 + (A - I)^3 - 6A$ is equal to

Let $ A $ be a $ 3 \times 3 $ matrix such that $ | \text{adj} (\text{adj} A) | = 81.
$ If $ S = \left\{ n \in \mathbb{Z}: \left| \text{adj} (\text{adj} A) \right|^{\frac{(n - 1)^2}{2}} = |A|^{(3n^2 - 5n - 4)} \right\}, $ then the value of $ \sum_{n \in S} |A| (n^2 + n) $ is:

Let $ A $ be a set defined as $ A = \{ 2, 3, 6, 9 \} $. Find the number of singular matrices of order $ 2 \times 2 $ such that elements are from the set $ A $.

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:

If $A$ is a square matrix of order 3 such that \[ \det(A) = 3 \] and \[ \det(\text{adj}(-4 \, \text{adj}(-3 \, \text{adj}(3 \, \text{adj}((2A)^{-1}))))) = 2^{m^3 n}, \] then $m + 2n$ 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 $A = I_2 - MM^T$, where $M$ is a real matrix of order $2 \times 1$ such that the relation $M^T M = I_1$ holds. If $\lambda$ is a real number such that the relation $AX = \lambda X$ holds for some non-zero real matrix $X$ of order $2 \times 1$, then the sum of squares of all possible values of $\lambda$ is equal to:

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: