Matrices & Determinants · JEE Main previous year questions

2026 19 questions

1. Q42026 · 8 Apr 2026 · Shift 2

   Let $\mathrm{A}=\left[\begin{array}{ccc}\alpha & 1& 2\\ 2& 3& 0\\ 0& 4& 5\end{array}\right]$ and $\mathrm{B}=\left[\begin{array}{ccc}1& 0& 0\\ 0& -5\alpha & 0\\ 0& 4\alpha & -2\alpha \end{array}\right]+\mathrm{adj}(\mathrm{A})$. If $\det (\mathrm{B})=66$, then $\det (\mathrm{adj}(\mathrm{A}))$ equals :

   1. A.$289$
   2. B.$361$
   3. C.$441$
   4. D.$529$
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   Answer: C

2. Q212026 · 6 Apr 2026 · Shift 1· Numerical answer

   Let $\mathrm{A}=\left[\begin{array}{ccc}-1& 1& -1\\ 1& 0& 1\\ 0& 0& 1\end{array}\right]$ satisfy ${\mathrm{A}}^2+\alpha (\mathrm{adj}(\mathrm{adj}(\mathrm{A})))+\beta (\mathrm{adj}(\mathrm{A})(\mathrm{adj}(\mathrm{adj}(\mathrm{A}))))=\left[\begin{array}{ccc}2& -2& 2\\ -2& 0& -1\\ 0& 0& -1\end{array}\right]$ for some $\alpha ,\beta \in \mathbb{R}$. Then $(\alpha -\beta {)}^2$ is equal to ____.

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   Answer: 4

3. Q42026 · 6 Apr 2026 · Shift 2

   The sum of all possible values of $\theta \in [0,2\pi ]$, for which the system of equations : $x\cos 3\theta -8y-12z=0$ $x\cos 2\theta +3y+3z=0$ $x+y+3z=0$ has a non-trivial solution, is equal to :

   1. A.$\pi$
   2. B.$2\pi$
   3. C.$3\pi$
   4. D.$4\pi$
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   Answer: D

4. Q172026 · 6 Apr 2026 · Shift 2

   Let $\mathrm{A}=\left[\begin{array}{ccc}1& 3& -1\\ 2& 1& \alpha \\ 0& 1& -1\end{array}\right]$ be a singular matrix. Let $f(x)={\int }_0^x(t^2+2t+3)dt$, $x\in [1,\alpha ]$. If $\mathrm{M}$ and $\mathrm{m}$ are respectively the maximum and the minimum values of $f$ in $[1,\alpha ]$, then $3(\mathrm{M}-\mathrm{m})$ is equal to :

   1. A.$64$
   2. B.$68$
   3. C.$72$
   4. D.$76$
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   Answer: B

5. Q32026 · 5 Apr 2026 · Shift 1

   Let A be a $3\times 3$ matrix such that $${\mathrm{A}}^{\mathrm{T}}\left[\begin{array}{c}1\\ 0\\ 1\end{array}\right]=\left[\begin{array}{c}5\\ 2\\ 2\end{array}\right],{\mathrm{A}}^{\mathrm{T}}\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]=\left[\begin{array}{c}3\\ 1\\ 1\end{array}\right],\mathrm{A}\left[\begin{array}{c}1\\ 0\\ 1\end{array}\right]=\left[\begin{array}{c}3\\ 4\\ 4\end{array}\right]\text{ and }\mathrm{A}\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]=\left[\begin{array}{c}1\\ 3\\ 1\end{array}\right].$$ If $\det (\mathrm{A})=1$, then $\det (\mathrm{adj}({\mathrm{A}}^2+\mathrm{A}))$ is equal to:

   1. A.16
   2. B.25
   3. C.49
   4. D.64
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   Answer: D

6. Q42026 · 5 Apr 2026 · Shift 2

   Let $\mathrm{M}$ be a $3\times 3$ matrix such that $$\mathrm{M}\left(\begin{array}{c}1\\ 0\\ 0\end{array}\right)=\left(\begin{array}{c}1\\ 2\\ 3\end{array}\right),\mathrm{M}\left(\begin{array}{c}0\\ 1\\ 0\end{array}\right)=\left(\begin{array}{c}0\\ 1\\ 2\end{array}\right)\text{ and }\mathrm{M}\left(\begin{array}{c}0\\ 0\\ 1\end{array}\right)=\left(\begin{array}{c}-1\\ 1\\ 1\end{array}\right).\text{ If }\mathrm{M}\left(\begin{array}{c}x\\ y\\ z\end{array}\right)=\left(\begin{array}{c}1\\ 7\\ 11\end{array}\right),\text{ then }x+y+z\text{ equals :}$$

   1. A.$4$
   2. B.$5$
   3. C.$7$
   4. D.$11$
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   Answer: B

7. Q52026 · 4 Apr 2026 · Shift 1

   Let $S=\left\{A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]:a,b,c,d\in \{0,1,2,3,4\}\text{ and }{A}^2-4A+3I=0\right\}$ 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. A.20
   2. B.17
   3. C.21
   4. D.19
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   Answer: D

8. Q62026 · 4 Apr 2026 · Shift 1

   Let $A=\left[\begin{array}{ccc}1& 1& 2\\ -2& 0& 1\\ 1& 3& 5\end{array}\right]$. Then the sum of all elements of the matrix $\mathrm{adj}(\mathrm{adj}(2(\mathrm{adj}A{)}^{-1}))$ is equal to:

   1. A.3
   2. B.4
   3. C.$-4$
   4. D.$-3$
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   Answer: D

9. Q32026 · 2 Apr 2026 · Shift 2

   If the system of equations $$\begin{array}{rl}x+5y+6z& =4,\\ 2x+3y+4z& =7,\\ x+6y+az& =b\end{array}$$ has infinitely many solutions, then the point (a, b) lies on the line

   1. A.$y-x=3$
   2. B.$x-y=3$
   3. C.$x+y=11$
   4. D.$x+y=12$
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   Answer: B

10. Q222026 · 2 Apr 2026 · Shift 2· Numerical answer

    Consider the matrices $\mathrm{A}=\left[\begin{array}{cc}2& -2\\ 4& -2\end{array}\right]$ and $\mathrm{B}=\left[\begin{array}{cc}3& 9\\ 1& 3\end{array}\right]$. If matrices P and Q are such that $\mathrm{PA}=\mathrm{B}$ and $\mathrm{AQ}=\mathrm{B}$, then the absolute value of the sum of the diagonal elements of $2(\mathrm{P}+\mathrm{Q})$ is ____.

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    Answer: 34

11. Q82026 · 28 Jan 2026 · Shift 1

    Let $A,B$ and $C$ be three $2\times 2$ matrices with real entries such that $B=(I+A{)}^{-1}$ and $\mathrm{A}+\mathrm{C}=\mathrm{I}$. If $\mathrm{BC}=\left[\begin{array}{cc}1& -5\\ -1& 2\end{array}\right]$ and $\mathrm{CB}\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]=\left[\begin{array}{c}12\\ -6\end{array}\right]$, then $x_1+x_2$ is

    1. A.4
    2. B.2
    3. C.0
    4. D.-2
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    Answer: C

12. Q242026 · 28 Jan 2026 · Shift 2· Numerical answer

    Let $A=\left[\begin{array}{cc}3& -4\\ 1& -1\end{array}\right]$ and $B$ be two matrices such that $A^{100}=100B+I$. Then the sum of all the elements of $B^{100}$ is ________

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    Answer: 0

13. Q232026 · 24 Jan 2026 · Shift 1· Numerical answer

    The number of $3\times 2$ 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 ${\mathrm{A}}^{\mathrm{T}}\mathrm{A}$ is 5 , is ____

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    Answer: 312

14. Q212026 · 23 Jan 2026 · Shift 1· Numerical answer

    Let $|A|=6$, where $A$ is a $3\times 3$ matrix. If $|\mathrm{adj}(3\mathrm{adj}(A^2\cdot \mathrm{adj}(2A)))|=2^m\cdot 3^n$, $m,n\in \mathbf{N}$, then $m+n$ is equal to ____ .

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    Answer: 62

15. Q252026 · 23 Jan 2026 · Shift 2· Numerical answer

    Let $A=\left[\begin{array}{ccc}0& 2& -3\\ -2& 0& 1\\ 3& -1& 0\end{array}\right]$ 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 _____.

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    Answer: 3

16. Q92026 · 22 Jan 2026 · Shift 1

    If $\mathrm{A}=\left[\begin{array}{cc}2& 3\\ 3& 5\end{array}\right]$, then the determinant of the matrix $\left({\mathrm{A}}^{2025}-3{\mathrm{A}}^{2024}+{\mathrm{A}}^{2023}\right)$ is

    1. A.12
    2. B.24
    3. C.28
    4. D.16
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    Answer: D

17. Q222026 · 22 Jan 2026 · Shift 1· Numerical answer

    Let A be a $3\times 3$ matrix such that $\mathrm{A}+{\mathrm{A}}^{\mathrm{T}}=\mathrm{O}$. If $\mathrm{A}\left[\begin{array}{c}1\\ -1\\ 0\end{array}\right]=\left[\begin{array}{c}3\\ 3\\ 2\end{array}\right],{\mathrm{A}}^2\left[\begin{array}{c}1\\ -1\\ 0\end{array}\right]=\left[\begin{array}{c}-3\\ 19\\ -24\end{array}\right]$ and $\det (\mathrm{adj}(2\mathrm{adj}(\mathrm{A}+\mathrm{I})))=(2{)}^{\alpha }\cdot (3{)}^{\beta }\cdot (11{)}^{\gamma },\alpha ,\beta ,\gamma$ are non-negative integers, then $\alpha +\beta +\gamma$ is equal to ____

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    Answer: 18

18. Q252026 · 21 Jan 2026 · Shift 1· Numerical answer

    For some $\alpha ,\beta \in \mathbf{R}$, let $A=\left[\begin{array}{cc}\alpha & 2\\ 1& 2\end{array}\right]$ and $B=\left[\begin{array}{cc}1& 1\\ 1& \beta \end{array}\right]$ be such that $A^2-4A+2I=B^2-3B+I=O$. Then ${\left(\det \left(\mathrm{adj}\left(A^3-B^3\right)\right)\right)}^2$ is equal to ____ .

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    Answer: 225

19. Q162026 · 21 Jan 2026 · Shift 2

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

    1. A.$x=16,y=3$
    2. B.$x=5,y=7$
    3. C.$x=11,y=2$
    4. D.$x=18,y=11$
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    Answer: C