Skip to main content
Yukti Vault

Algebra

Matrices: JEE Main & Advanced PYQs

203 previous year questions on Matrices from JEE Main and JEE Advanced papers, each with its exam and year, an answer and a solution.

Practice this chapter

Questions 1-50 of 203

  1. If $A=\begin{pmatrix}0&\sin\alpha\\ \sin\alpha&0\end{pmatrix}$ and $\det\left(A^{2}-\frac{1}{2}I\right)=0$, then a possible value of $\alpha$ is :
  2. Let $A$ be a symmetric matrix of order $2$ with integer entries. If the sum of the diagonal elements of $A^{2}$ is $1$, then the possible number of such matrices is :
  3. Let $A=[a_{ij}]$ be a real matrix of order $3\times3$, such that $a_{i1}+a_{i2}+a_{i3}=1$, for $i=1,2,3$. Then, the sum of all the entries of the matrix $A^{3}$ is equal …
  4. Let $A=\begin{bmatrix}\alpha&1&2\\2&3&0\\0&4&5\end{bmatrix}$ and $B=\begin{bmatrix}1&0&0\\0&-5\alpha&0\\0&4\alpha&-2\alpha\end{bmatrix}+\operatorname{adj}(A)$. If …
  5. Let $A+2B=\begin{bmatrix}1&2&0\\6&-3&3\\-5&3&1\end{bmatrix}$ and $2A-B=\begin{bmatrix}2&-1&5\\2&-1&6\\0&1&2\end{bmatrix}$. If $\operatorname{Tr}(A)$ denotes the sum of …
  6. Let $A$ and $B$ be $3\times3$ real matrices such that $A$ is symmetric matrix and $B$ is skew-symmetric matrix. Then the system of linear equations …
  7. Let $M$ be a $3\times3$ matrix such that $M\begin{pmatrix}1\\0\\0\end{pmatrix}=\begin{pmatrix}1\\2\\3\end{pmatrix}$ …
  8. Let $A=\begin{bmatrix}1&0&0\\3&1&0\\9&3&1\end{bmatrix}$ and $B=[b_{ij}]$, $1\le i,j\le3$. If $B=A^{99}-I$, then the value of $\frac{b_{31}-b_{21}}{b_{32}}$ is :
  9. If for the matrix, $A=\begin{bmatrix}1&-\alpha\\ \alpha&\beta\end{bmatrix}$, $AA^{T}=I_2$, then the value of $\alpha^{4}+\beta^{4}$ is :
  10. If $P=\begin{bmatrix}1&0\\\frac{1}{2}&1\end{bmatrix}$, then $P^{50}$ is :
  11. Let $A=\begin{bmatrix}0&-2\\2&0\end{bmatrix}$. If $M$ and $N$ are two matrices given by $M=\sum_{k=1}^{10}A^{2k}$ and $N=\sum_{k=1}^{10}A^{2k-1}$, then $MN^{2}$ is :
  12. Let $A=\begin{bmatrix}a_1\\a_2\end{bmatrix}$ and $B=\begin{bmatrix}b_1\\b_2\end{bmatrix}$ be two $2\times1$ matrices with real entries such that $A=XB$, where …
  13. Let $A=[a_{ij}]_{2\times2}$, where $a_{ij}\neq0$ for all $i,j$ and $A^{2}=I$. Let $a$ be the sum of all diagonal elements of $A$ and $b=|A|$. Then $3a^{2}+4b^{2}$ is …
  14. Let $A=\begin{bmatrix}1&\frac{1}{51}\\0&1\end{bmatrix}$. If $B=\begin{bmatrix}1&2\\-1&-1\end{bmatrix}A\begin{bmatrix}-1&-2\\1&1\end{bmatrix}$, then the sum of all the …
  15. Let $A=\begin{pmatrix}1&2\\-2&-5\end{pmatrix}$. Let $\alpha,\beta\in\mathbb{R}$ be such that $\alpha A^2+\beta A=2I$. Then $\alpha+\beta$ is equal to
  16. Let the matrix $A=\begin{bmatrix}0&1&0\\0&0&1\\1&0&0\end{bmatrix}$ and the matrix $B_0=A^{49}+2A^{98}$. If $B_n=\mathrm{Adj}(B_{n-1})$ for all $n\ge1$, then $\det(B_4)$ …
  17. Let $A$ be a square matrix such that $AA^{T}=I$. Then $\frac{1}{2}A\left[\left(A+A^{T}\right)^{2}+\left(A-A^{T}\right)^{2}\right]$ is equal to
  18. Let $A$ and $B$ be any two $3\times3$ symmetric and skew symmetric matrices respectively. Then which of the following is NOT true?
  19. Let $P=\begin{bmatrix}3&-1&-2\\2&0&\alpha\\3&-5&0\end{bmatrix}$, where $\alpha\in\mathbf{R}$. Suppose $Q=[q_{ij}]$ is a matrix satisfying $PQ=kI_3$ for some non-zero …
  20. Let $A$ be a $3\times3$ matrix such that $\mathrm{adj}\,A=\begin{bmatrix}2&-1&1\\-1&0&2\\1&-2&-1\end{bmatrix}$ and $B=\mathrm{adj}(\mathrm{adj}\,A)$. If $|A|=\lambda$ …
  21. Let $A$ be a $3\times3$ real matrix such that $A\begin{pmatrix}1\\0\\1\end{pmatrix}=2\begin{pmatrix}1\\0\\1\end{pmatrix}$ …
  22. Let $A=\begin{bmatrix}\alpha&-1\\6&\beta\end{bmatrix}$, $\alpha>0$, such that $\det(A)=0$ and $\alpha+\beta=1$. If $I$ denotes $2\times2$ identity matrix, then the …
  23. Let $A=[a_{ij}]$ be a square matrix of order $3$ such that $a_{ij}=2^{j-i}$, for all $i,j=1,2,3$. Then, the matrix $A^{2}+A^{3}+\ldots+A^{10}$ is equal to :
  24. If $A=\begin{bmatrix}\cos\theta&i\sin\theta\\i\sin\theta&\cos\theta\end{bmatrix}$, $\left(\theta=\frac\pi{24}\right)$ and $A^5=\begin{bmatrix}a&b\\c&d\end{bmatrix}$ …
  25. If $A=\begin{pmatrix}\frac{1}{\sqrt{5}}&\frac{2}{\sqrt{5}}\\-\frac{2}{\sqrt{5}}&\frac{1}{\sqrt{5}}\end{pmatrix}$, $B=\begin{pmatrix}1&0\\i&1\end{pmatrix}$ …
  26. Let $a,b,c\in\mathbf{R}$ be all non-zero and satisfy $a^3+b^3+c^3=2$. If the matrix $A=\begin{pmatrix}a&b&c\\b&c&a\\c&a&b\end{pmatrix}$ satisfies $A^TA=I$, then a value …
  27. Let $A=\begin{pmatrix}2&-1\\0&2\end{pmatrix}$. If $B=I-{}^{5}C_1(\operatorname{adj}A)+{}^{5}C_2(\operatorname{adj}A)^{2}-\ldots-{}^{5}C_5(\operatorname{adj}A)^{5}$, then …
  28. Let $A=\begin{bmatrix}x&y&z\\y&z&x\\z&x&y\end{bmatrix}$, where $x,y$ and $z$ are real numbers such that $x+y+z>0$ and $xyz=2$. If $A^{2}=I_3$, then the value of …
  29. Let $B=\begin{bmatrix}1&3&\alpha\\1&2&3\\\alpha&\alpha&4\end{bmatrix}$, $\alpha>2$ be the adjoint of a matrix $A$ and $|A|=2$. Then …
  30. Let $P$ be a square matrix such that $P^{2}=I-P$. For $\alpha,\beta,\gamma,\delta\in\mathbb{N}$, if $P^{\alpha}+P^{\beta}=\gamma I-29P$ and …
  31. If $A=\begin{bmatrix}0&-\tan\left(\frac{\theta}{2}\right)\\ \tan\left(\frac{\theta}{2}\right)&0\end{bmatrix}$ and …
  32. Let $\alpha$ be a solution of $x^{2}+x+1=0$, and for some $a$ and $b$ in $\mathbb{R}$ …
  33. Let $A=\begin{bmatrix}2&a&0\\1&3&1\\0&5&b\end{bmatrix}$. If $A^{3}=4A^{2}-A-21I$, where $I$ is the identity matrix of order $3\times3$, then $2a+3b$ is equal to
  34. Let $A$ be a matrix and $\begin{bmatrix}1&2\\0&1\end{bmatrix}A\begin{bmatrix}2&0\\-1&1\end{bmatrix}=\begin{bmatrix}1&0\\0&1\end{bmatrix}$. If $|A|$ and $|A^2|$ are the …
  35. Let $A$ be a $2\times2$ matrix with real entries such that $A'=\alpha A+I$, where $\alpha\in\mathbb{R}-\{-1,1\}$. If $\det\left(A^{2}-A\right)=4$, then the sum of all …
  36. Let $A=\begin{pmatrix}1&0&0\\0&1&1\\1&0&0\end{pmatrix}$. Then $A^{2025}-A^{2020}$ is equal to :
  37. Let $A=\begin{bmatrix}a&b\\c&d\end{bmatrix}$ and $B=\begin{bmatrix}\alpha\\ \beta\end{bmatrix}\neq\begin{bmatrix}0\\0\end{bmatrix}$ such that $AB=B$ and $a+d=2021$, then …
  38. If the matrix $A=\begin{pmatrix}0&2\\K&-1\end{pmatrix}$ satisfies $A(A^{3}+3I)=2I$, then the value of $K$ is :
  39. The number of symmetric matrices of order $3$, with all the entries from the set $\{0,1,2,3,4,5,6,7,8,9\}$, is
  40. Let $A$ be a $2\times2$ real matrix with entries from $\{0,1\}$ and $|A|\ne0$. Consider the following two statements : (P) If $A\ne I_2$, then $|A|=-1$ (Q) If $|A|=1$ …
  41. Let $P=\begin{bmatrix}\frac{\sqrt{3}}{2}&\frac{1}{2}\\-\frac{1}{2}&\frac{\sqrt{3}}{2}\end{bmatrix}$, $A=\begin{bmatrix}1&1\\0&1\end{bmatrix}$ and $Q=PAP^{T}$. If …
  42. If the matrix $A=\begin{bmatrix}1&0&0\\0&2&0\\3&0&-1\end{bmatrix}$ satisfies the equation $A^{20}+\alpha A^{19}+\beta A=\begin{bmatrix}1&0&0\\0&4&0\\0&0&1\end{bmatrix}$ …
  43. Consider the matrices $A=\begin{bmatrix}2&-2\\4&-2\end{bmatrix}$ and $B=\begin{bmatrix}3&9\\1&3\end{bmatrix}$. If matrices $P$ and $Q$ are such that $PA=B$ and $AQ=B$ …
  44. Let $A$ be a $2\times2$ matrix with $\det(A)=-1$ and $\det((A+I)(\mathrm{Adj}(A)+I))=4$. Then the sum of the diagonal elements of $A$ can be :
  45. If $A=\begin{bmatrix}1&5\\\lambda&10\end{bmatrix}$, $A^{-1}=\alpha A+\beta I$ and $\alpha+\beta=-2$, then $4\alpha^{2}+\beta^{2}+\lambda^{2}$ is equal to :
  46. Let $A$ and $B$ be two $3\times3$ non-zero real matrices such that $AB$ is a zero matrix. Then
  47. 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 …
  48. Which of the following matrices can NOT be obtained from the matrix $\begin{bmatrix}-1&2\\1&-1\end{bmatrix}$ by a single elementary row operation ?
  49. Let $R=\begin{pmatrix}x&0&0\\0&y&0\\0&0&z\end{pmatrix}$ be a non-zero $3\times3$ matrix, where …
  50. Let $A=\begin{pmatrix}0&1&0\\0&0&1\\1&0&0\end{pmatrix}$ and $B=\begin{pmatrix}1&1&1\\1&1&1\\1&1&1\end{pmatrix}$. Then the set …

Other chapters