Matrices & Determinants

33 JEE Main questions · Maths

MathsJEE Main 21 Jan 2026, Shift 1·Q24·Numerical·Moderate
Open
Adjoint and determinant properties
For some $\alpha, \beta \in \mathbb{R}$ , let $A = \begin{bmatrix} \alpha & 2\\ 1 & 2\end{bmatrix} \quad\text{and}\quad B = \begin{bmatrix} 1 & 1\\ 1 & \beta\end{bmatrix}$ be such that $A^2 - 4A + 2I = B^2 - 3B + I = O$ . Then $(\det(\operatorname{adj}(A^3 - B^3)))^2$ is equal to .
MathsJEE Main 21 Jan 2026, Shift 2·Q14·MCQ·Moderate
Open
Powers of a matrix
For the matrices $A = \begin{bmatrix} 3 & -4\\ 1 & -1\end{bmatrix} \quad\text{and}\quad B = \begin{bmatrix} -29 & 49\\ -13 & 18\end{bmatrix},$ if $(A^{15} + B)\begin{bmatrix} x\\ y\end{bmatrix} = \begin{bmatrix} 0\\ 0\end{bmatrix}$ , then among the following which one is true?
1. (1) $x = 5$ , $y = 7$
2. (2) $x = 18$ , $y = 11$
3. (3) $x = 11$ , $y = 2$
4. (4) $x = 16$ , $y = 3$
MathsJEE Main 21 Jan 2026, Shift 2·Q18·MCQ·Easy–moderate
Open
Consistency of linear equations
If the system of equations
$\begin{align*} 3x + y + 4z &= 3\\ 2x + \alpha y - z &= -3\\ x + 2y + z &= 4 \end{align*}$
has no solution, then the value of $\alpha$ is equal to:
1. (1)19
2. (2)4
3. (3)13
4. (4)23
We corrected a printing error in the original paper
Source printed the third equation as “ $X + 2y + z = 4$ ” (capital $X$ for $x$ ).
MathsJEE Main 22 Jan 2026, Shift 1·Q13·MCQ·Easy–moderate
Open
Determinant of matrix polynomial
If $A = \begin{bmatrix} 2 & 3\\ 3 & 5\end{bmatrix}$ , then the determinant of the matrix $(A^{2025} - 3A^{2024} + A^{2023})$ is
1. (1)28
2. (2)12
3. (3)24
4. (4)16
MathsJEE Main 22 Jan 2026, Shift 1·Q22·Numerical·Tough
Open
Adjoint and determinant properties
Let $A$ be a $3 \times 3$ matrix such that $A + A^T = O$ . If $A\begin{bmatrix} 1\\ -1\\ 0\end{bmatrix} = \begin{bmatrix} 3\\ 3\\ 2\end{bmatrix},\quad A^2\begin{bmatrix} 1\\ -1\\ 0\end{bmatrix} = \begin{bmatrix} -3\\ 19\\ -24\end{bmatrix}$ and $\det(\operatorname{adj}(2\operatorname{adj}(A + I))) = (2)^{\alpha}\cdot(3)^{\beta}\cdot(11)^{\gamma}$ , $\alpha, \beta, \gamma$ are non-negative integers, then $\alpha + \beta + \gamma$ is equal to .
MathsJEE Main 22 Jan 2026, Shift 2·Q1·MCQ·Moderate
Open
Consistency of linear equations
Let $n$ be the number obtained on rolling a fair die. If the probability that the system
$\begin{align*} x - ny + z &= 6\\ x + (n-2)y + (n+1)z &= 8\\ (n-1)y + z &= 1 \end{align*}$
Has a unique solution is $\dfrac{k}{6}$ , then the sum of $k$ and all possible values of $n$ is:
1. (1)21
2. (2)24
3. (3)20
4. (4)22
MathsJEE Main 22 Jan 2026, Shift 2·Q9·MCQ·Moderate
Open
Adjoint and inverse of matrix
If $X = \begin{bmatrix} x\\ y\\ z\end{bmatrix}$ is a solution of the system of equations $AX = B$ , where $\operatorname{adj} A = \begin{bmatrix} 4 & 2 & 2\\ -5 & 0 & 5\\ 1 & -2 & 3\end{bmatrix} \quad\text{and}\quad B = \begin{bmatrix} 4\\ 0\\ 2\end{bmatrix},$ then $\lvert x + y + z\rvert$ is equal to:
1. (1)3
2. (2) $\dfrac32$
3. (3)1
4. (4)2
MathsJEE Main 23 Jan 2026, Shift 1·Q11·MCQ·Moderate
Open
Evaluation of determinants
Among the statements:
I: If $\begin{vmatrix} 1 & \cos\alpha & \cos\beta\\ \cos\alpha & 1 & \cos\gamma\\ \cos\beta & \cos\gamma & 1\end{vmatrix} = \begin{vmatrix} 0 & \cos\alpha & \cos\beta\\ \cos\alpha & 0 & \cos\gamma\\ \cos\beta & \cos\gamma & 0\end{vmatrix}$ , then $\cos^2\alpha + \cos^2\beta + \cos^2\gamma = \dfrac32$ , and
II: If $\begin{vmatrix} x^2 + x & x + 1 & x - 2\\ 2x^2 + 3x - 1 & 3x & 3x - 3\\ x^2 + 2x + 3 & 2x - 1 & 2x - 1\end{vmatrix} = px + q$ , then $p^2 = 196q^2$ ,
1. (1)both are false
2. (2)only II is true
3. (3)both are true
4. (4)only I is true
MathsJEE Main 23 Jan 2026, Shift 1·Q21·Numerical·Moderate
Open
Adjoint and determinant properties
Let $\lvert A\rvert = 6$ , where $A$ is a $3\times3$ matrix. If $\lvert\operatorname{adj}(3\operatorname{adj}(A^2\cdot\operatorname{adj}(2A)))\rvert = 2^m\cdot3^n$ , $m, n \in \mathbb{N}$ , then $m + n$ is equal to .
MathsJEE Main 23 Jan 2026, Shift 2·Q6·MCQ·Easy–moderate
Open
Consistency of linear equations
The system of linear equations
$\begin{align*} x + y + z &= 6\\ 2x + 5y + az &= 36\\ x + 2y + 3z &= b \end{align*}$
has
1. (1)unique solution for $a = 8$ and $b = 16$
2. (2)infinitely many solutions for $a = 8$ and $b = 14$
3. (3)infinitely many solutions for $a = 8$ and $b = 16$
4. (4)unique solution for $a = 8$ and $b = 14$
MathsJEE Main 23 Jan 2026, Shift 2·Q24·Numerical·Moderate
Open
Orthogonal and skew-symmetric matrices
Let $A = \begin{bmatrix} 0 & 2 & -3\\ -2 & 0 & 1\\ 3 & -1 & 0\end{bmatrix}$ and $B$ be a matrix such that $B(I - A) = I + A$ . Then the sum of the diagonal elements of $B^{T}B$ is equal to .
MathsJEE Main 24 Jan 2026, Shift 1·Q25·Numerical·Moderate
Open
Trace of transpose product
The number of $3 \times 2$ matrices $A$ , which can be formed using the elements of the set $\{-2, -1, 0,\allowbreak 1, 2\}$ such that the sum of all the diagonal elements of $A^{T}A$ is 5, is .
MathsJEE Main 24 Jan 2026, Shift 2·Q1·MCQ·Moderate
Open
Adjoint and determinant properties
Let $f(x) = \displaystyle\int\dfrac{7x^{10} + 9x^8}{(1 + x^2 + 2x^9)^2}\,dx$ , $x > 0$ , $\displaystyle\lim_{x\to0} f(x) = 0$ and $f(1) = \dfrac14$ . If $A = \begin{bmatrix} 0 & 0 & 1\\[2pt] \dfrac14 & f'(1) & 1\\[2pt] \alpha^2 & 4 & 1\end{bmatrix}$ and $B = \operatorname{adj}(\operatorname{adj} A)$ be such that $\lvert B\rvert = 81$ , then $\alpha^2$ is equal to
1. (1)2
2. (2)3
3. (3)1
4. (4)4
MathsJEE Main 24 Jan 2026, Shift 2·Q8·MCQ·Moderate
Open
Determinant and adjoint properties
Let $P = [p_{ij}]$ and $Q = [q_{ij}]$ be two square matrices of order 3 such that $q_{ij} = 2^{(i + j - 1)}p_{ij}$ and $\det(Q) = 2^{10}$ . Then the value of $\det(\operatorname{adj}(\operatorname{adj} P))$ is:
1. (1)32
2. (2)16
3. (3)81
4. (4)124
MathsJEE Main 28 Jan 2026, Shift 1·Q16·MCQ·Moderate
Open
Inverse of a matrix
Let $A$ , $B$ and $C$ be three $2\times2$ matrices with real entries such that $B = (I + A)^{-1}$ and $A + C = I$ . If $BC = \begin{bmatrix} 1 & -5\\ -1 & 2\end{bmatrix} \quad\text{and}\quad CB\begin{bmatrix} x_1\\ x_2\end{bmatrix} = \begin{bmatrix} 12\\ -6\end{bmatrix},$ then $x_1 + x_2$ is
1. (1)2
2. (2)0
3. (3) $-2$
4. (4)4
MathsJEE Main 28 Jan 2026, Shift 2·Q25·Numerical·Moderate
Open
Powers of a nilpotent matrix
Let $A = \begin{bmatrix} 3 & -4\\ 1 & -1\end{bmatrix}$ and $B$ be two matrices such that $A^{100} = 100B + I$ . Then the sum of all the elements of $B^{100}$ is .
MathsJEE Main 2 Apr 2026, Shift 1·Q3·MCQ·Easy–moderate
Open
Consistency of linear systems
Let $\alpha, \beta \in \mathbb{R}$ be such that the system of linear equations
$\begin{align*} x + 2y + z &= 5\\ 2x + y + \alpha z &= 5\\ 8x + 4y + \beta z &= 18 \end{align*}$
has no solution. Then $\dfrac{\beta}{\alpha}$ is equal to:
1. (1) $-4$
2. (2)4
3. (3)8
4. (4) $-8$
MathsJEE Main 2 Apr 2026, Shift 1·Q4·MCQ·Moderate
Open
Adjoint and determinant properties
Let $A = \begin{bmatrix} 1 & 2\\ 1 & \alpha\end{bmatrix}$ and $B = \begin{bmatrix} 3 & 3\\ \beta & 2\end{bmatrix}$ . If $A^2 - 4A + I = O$ and $B^2 - 5B - 6I = O$ , then among the two statements:
- (S1): $[(B - A)(B + A)]^{T} = \begin{bmatrix} 13 & 15\\ 7 & 10\end{bmatrix}$
and
- (S2): $\det(\operatorname{adj}(A + B)) = -5$ ,
1. (1)only (S1) is correct
2. (2)only (S2) is correct
3. (3)both (S1) and (S2) are correct
4. (4)both (S1) and (S2) are wrong
MathsJEE Main 2 Apr 2026, Shift 2·Q3·MCQ·Moderate
Open
Consistency of linear equations
If the system of equations
$\begin{align*} x + 5y + 6z &= 4,\\ 2x + 3y + 4z &= 7,\\ x + 6y + az &= b \end{align*}$
has infinitely many solutions, then the point $(a, b)$ lies on the line
1. (1) $y - x = 3$
2. (2) $x - y = 3$
3. (3) $x + y = 11$
4. (4) $x + y = 12$
MathsJEE Main 2 Apr 2026, Shift 2·Q22·Numerical·Easy–moderate
Open
Inverse of a matrix
Consider the matrices $A = \begin{bmatrix} 2 & -2\\ 4 & -2\end{bmatrix} \quad\text{and}\quad B = \begin{bmatrix} 3 & 9\\ 1 & 3\end{bmatrix}.$ If matrices $P$ and $Q$ are such that $PA = B$ and $AQ = B$ , then the absolute value of the sum of the diagonal elements of $2(P + Q)$ is .
We corrected a printing error in the original paper
Source printed “B” directly followed by the matrix, without the “=” sign.
MathsJEE Main 4 Apr 2026, Shift 1·Q5·MCQ·Moderate
Open
Matrix equation with integer entries
Let $S = \left\{A = \begin{bmatrix} a & b\\ c & d\end{bmatrix} : a, b, c, d \in \{0, 1, 2, 3, 4\} \text{ and } A^2 - 4A + 3I = 0\right\}$ be a set of $2 \times 2$ matrices. Then the number of matrices in $S$ , for which the sum of the diagonal elements is equal to 4, is:
1. (1)20
2. (2)17
3. (3)21
4. (4)19
MathsJEE Main 4 Apr 2026, Shift 1·Q6·MCQ·Moderate
Open
Adjoint and inverse properties
Let $A = \begin{bmatrix} 1 & 1 & 2\\ -2 & 0 & 1\\ 1 & 3 & 5\end{bmatrix}.$ Then the sum of all elements of the matrix $\operatorname{adj}(\operatorname{adj}(2(\operatorname{adj}A)^{-1}))$ is equal to:
1. (1)3
2. (2)4
3. (3) $-4$
4. (4) $-3$
MathsJEE Main 4 Apr 2026, Shift 2·Q3·MCQ·Easy–moderate
Open
System of linear equations
If the system of equations: $\begin{aligned} x + y + z &= 5\\ x + 2y + 3z &= 9\\ x + 3y + \lambda z &= \mu\end{aligned}$ has infinitely many solutions, then the value of $\lambda + \mu$ is:
1. (1)16
2. (2)18
3. (3)19
4. (4)21
MathsJEE Main 4 Apr 2026, Shift 2·Q6·MCQ·Moderate
Open
Characteristic equation of a matrix
Let $A = \begin{bmatrix} 1 & 2 & 7\\ 4 & -2 & 8\\ 3 & 8 & -7\end{bmatrix}$ and $\det(A - \alpha I) = 0$ , where $\alpha$ is a real number. If the largest possible value of $\alpha$ is $p$ , then the circle $(x-p)^2 + (y-2p)^2 = 320$ , intersects the co-ordinate axes at
1. (1)1 point
2. (2)2 points
3. (3)3 points
4. (4)4 points
MathsJEE Main 5 Apr 2026, Shift 1·Q3·MCQ·Moderate
Open
Determinant of adjoint
Let $A$ be a $3 \times 3$ matrix such that $A^{T}\begin{bmatrix} 1\\ 0\\ 1\end{bmatrix} = \begin{bmatrix} 5\\ 2\\ 2\end{bmatrix},\quad A^{T}\begin{bmatrix} 0\\ 0\\ 1\end{bmatrix} = \begin{bmatrix} 3\\ 1\\ 1\end{bmatrix},\quad A\begin{bmatrix} 1\\ 0\\ 1\end{bmatrix} = \begin{bmatrix} 3\\ 4\\ 4\end{bmatrix}\quad\text{and}\quad A\begin{bmatrix} 0\\ 0\\ 1\end{bmatrix} = \begin{bmatrix} 1\\ 3\\ 1\end{bmatrix}$ If $\det(A) = 1$ , then $\det(\operatorname{adj}(A^2 + A))$ is equal to:
1. (1)16
2. (2)25
3. (3)49
4. (4)64
MathsJEE Main 5 Apr 2026, Shift 1·Q4·MCQ·Moderate
Open
Homogeneous system infinitely many solutions
Consider the system of linear equations in $x, y, z$ : $\begin{aligned} &x + 2y + tz = 0,\\ &6x + y + 5tz = 0,\\ &3x + t^2y + f(t)z = 0, \end{aligned}$ Where $f : \mathbb{R} \to \mathbb{R}$ is a differentiable function. If this system has infinitely many solutions for all $t \in \mathbb{R}$ , then $f$
1. (1)is a constant function
2. (2)is strictly increasing on $\mathbb{R}$
3. (3)is strictly decreasing on $\mathbb{R}$
4. (4)has two critical points
MathsJEE Main 5 Apr 2026, Shift 2·Q3·MCQ·Moderate
Open
Determinant with summation of terms
If $f : \mathbb{N} \to \mathbb{Z}$ is defined by $f(n) = \begin{vmatrix} n & -1 & -5\\ -2n^2 & 3(2k+1) & 2k+1\\ -3n^3 & 3k(2k+1) & 3k(k+2)+1 \end{vmatrix},\ k \in \mathbb{N}, \text{ and}$ $\displaystyle\sum_{n=1}^{k} f(n) = 98$ , then $k$ is equal to:
1. (1)3
2. (2)4
3. (3)5
4. (4)6
MathsJEE Main 5 Apr 2026, Shift 2·Q4·MCQ·Easy–moderate
Open
Matrix from images of basis vectors
Let $M$ be a $3 \times 3$ matrix such that $M\begin{pmatrix}1\\0\\0\end{pmatrix} = \begin{pmatrix}1\\2\\3\end{pmatrix},\quad M\begin{pmatrix}0\\1\\0\end{pmatrix} = \begin{pmatrix}0\\1\\2\end{pmatrix} \quad\text{and}\quad M\begin{pmatrix}0\\0\\1\end{pmatrix} = \begin{pmatrix}-1\\1\\1\end{pmatrix}.$ If $M\begin{pmatrix}x\\y\\z\end{pmatrix} = \begin{pmatrix}1\\7\\11\end{pmatrix}$ , then $x + y + z$ equals:
1. (1)4
2. (2)5
3. (3)7
4. (4)11
We corrected a printing error in the original paper
Answer corrected; the source key is (3) — the columns of $M$ are $(1,2,3)$ , $(0,1,2)$ , $(-1,1,1)$ , so $x - z = 1$ , $2x + y + z = 7$ , $3x + 2y + z = 11$ give $(x,y,z) = (2,2,1)$ and $x + y + z = 5$ , option (2).
MathsJEE Main 6 Apr 2026, Shift 1·Q21·Numerical·Moderate
Open
Properties of adjoint matrix
Let $A = \begin{bmatrix} -1 & 1 & -1\\ 1 & 0 & 1\\ 0 & 0 & 1\end{bmatrix}$ satisfy $A^2 + \alpha(\operatorname{adj}(\operatorname{adj}(A))) + \beta(\operatorname{adj}(A)(\operatorname{adj}(\operatorname{adj}(A)))) = \begin{bmatrix} 2 & -2 & 2\\ -2 & 0 & -1\\ 0 & 0 & -1\end{bmatrix}$ for Some $\alpha, \beta \in \mathbb{R}$ . Then $(\alpha - \beta)^2$ is equal to .
MathsJEE Main 6 Apr 2026, Shift 2·Q4·MCQ·Moderate
Open
Homogeneous system non-trivial solution
The sum of all possible values of $\theta \in [0, 2\pi]$ , for which the system of equations:
$\begin{align*} x\cos3\theta - 8y - 12z &= 0\\ x\cos2\theta + 3y + 3z &= 0\\ x + y + 3z &= 0 \end{align*}$
has a non-trivial solution, is equal to:
1. (1) $\pi$
2. (2) $2\pi$
3. (3) $3\pi$
4. (4) $4\pi$
MathsJEE Main 6 Apr 2026, Shift 2·Q5·MCQ·Moderate
Open
Powers of a matrix
Let $A = \begin{bmatrix} 1 & 0 & 0\\ 3 & 1 & 0\\ 9 & 3 & 1\end{bmatrix} \quad\text{and}\quad B = [b_{ij}],\ 1 \le i, j \le 3.$ If $B = A^{99} - I$ , then the value of $\dfrac{b_{31} - b_{21}}{b_{32}}$ is:
1. (1)99
2. (2)199
3. (3)149
4. (4)159
We corrected a printing error in the original paper
Source printed “ $B = [b_{ij}]$ , $l \le i, j \le 3$ ” (letter l for the digit 1).
MathsJEE Main 8 Apr 2026, Shift 2·Q3·MCQ·Easy–moderate
Open
Consistency of linear equations
If the system of linear equations:
$\begin{align*} x + y + z &= 6,\\ x + 2y + 5z &= 10,\\ 2x + 3y + \lambda z &= \mu \end{align*}$
has infinitely many solutions, then the value of $\lambda + \mu$ equals.
1. (1)12
2. (2)16
3. (3)22
4. (4)28
MathsJEE Main 8 Apr 2026, Shift 2·Q4·MCQ·Moderate
Open
Adjoint and determinant properties
Let $A = \begin{bmatrix} \alpha & 1 & 2\\ 2 & 3 & 0\\ 0 & 4 & 5\end{bmatrix} \quad\text{and}\quad B = \begin{bmatrix} 1 & 0 & 0\\ 0 & -5\alpha & 0\\ 0 & 4\alpha & -2\alpha\end{bmatrix} + \operatorname{adj}(A).$ If $\det(B) = 66$ , then $\det(\operatorname{adj}(A))$ equals:
1. (1)289
2. (2)361
3. (3)441
4. (4)529