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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.

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Questions 101-150 of 203

  1. Let $A$ and $B$ be any two $3\times3$ matrices. If $A$ is symmetric and $B$ is skew symmetric, then the matrix $AB-BA$ is
  2. If $p,q,r$ are $3$ real numbers satisfying the matrix equation …
  3. Let $A$ be a $3\times3$ matrix such that $A\begin{bmatrix}1&2&3\\0&2&3\\0&1&1\end{bmatrix}=\begin{bmatrix}0&0&1\\1&0&0\\0&1&0\end{bmatrix}$. Then $A^{-1}$ is :
  4. Let $A$ and $B$ be two symmetric matrices of order $3$. Statement - 1: $A(BA)$ and $(AB)A$ are symmetric matrices. Statement - 2: $AB$ is symmetric matrix if matrix …
  5. Let $A$ be a $3\times3$ matrix, such that $A^{2}-5A+7I=O$. Statement - I : $A^{-1}=\frac17(5I-A)$. Statement - II : The polynomial $A^{3}-2A^{2}-3A+I$ can be reduced to …
  6. The total number of matrices $A=\begin{pmatrix}0&2y&1\\2x&y&-1\\2x&-y&1\end{pmatrix}$, $(x,y\in R,\ x\neq y)$ for which $A^{T}A=3I_3$ is :
  7. The matrix $A^{2}+4A-5I$, where $I$ is identity matrix and $A=\begin{bmatrix}1&2\\4&-3\end{bmatrix}$, equals :
  8. If $A=\begin{bmatrix}-4&-1\\3&1\end{bmatrix}$, then the determinant of the matrix $\left(A^{2016}-2A^{2015}-A^{2014}\right)$ is :
  9. Let $A$ be a $2\times2$ matrix with non-zero entries and let $A^{2}=I$, where $I$ is $2\times2$ identity matrix. Define $Tr(A)=$ sum of diagonal elements of $A$ and …
  10. If $A$ and $B$ are square matrices of size $n\times n$ such that $A^{2}-B^{2}=(A-B)(A+B)$, then which of the following will be always true?
  11. Let $S=\left\{\begin{pmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{pmatrix}:a_{ij}\in\{0,1,2\},\ a_{11}=a_{22}\right\}$. Then the number of non-singular matrices in the set …
  12. For two $3\times3$ matrices $A$ and $B$, let $A+B=2B'$ and $3A+2B=I_3$, where $B'$ is the transpose of $B$ and $I_3$ is $3\times3$ identity matrix. Then :
  13. Let $A=\begin{bmatrix}\cos\alpha&-\sin\alpha\\\sin\alpha&\cos\alpha\end{bmatrix}$, $\alpha\in R$ such that $A^{32}=\begin{bmatrix}0&-1\\1&0\end{bmatrix}$. Then, a value …
  14. If $M=\begin{pmatrix}\frac{5}{2}&\frac{3}{2}\\-\frac{3}{2}&-\frac{1}{2}\end{pmatrix}$, then which of the following matrices is equal to $M^{2022}$ ?
  15. Let $A=\begin{pmatrix}0&2q&r\\p&q&-r\\p&-q&r\end{pmatrix}$. If $AA^{T}=I_3$, then $|p|$ is :
  16. Let $P_1=I=\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}$, $P_2=\begin{bmatrix}1&0&0\\0&0&1\\0&1&0\end{bmatrix}$ …
  17. Consider the matrix $$P=\begin{pmatrix}2&0&0\\0&2&0\\0&0&3\end{pmatrix}.$$ Let the transpose of a matrix $X$ be denoted by $X^T$. Then the number of $3\times3$ …
  18. Let $\alpha$ and $\beta$ be the distinct roots of the equation $x^2+x-1=0$. Consider the set $T=\{1,\alpha,\beta\}$. For a $3\times3$ matrix $M=(a_{ij})_{3\times3}$ …
  19. Which one of the following matrices can be obtained by performing elementary row transformations on the $3\times3$ identity matrix ?
  20. Let $\alpha$ be a root of the equation $x^{2}+x+1=0$ and the matrix $A=\frac{1}{\sqrt3}\begin{bmatrix}1&1&1\\1&\alpha&\alpha^{2}\\1&\alpha^{2}&\alpha^{4}\end{bmatrix}$ …
  21. Let $A=[a_{ij}]$ be a matrix of order $3\times3$, with $a_{ij}=(\sqrt2)^{i+j}$. If the sum of all the elements in the third row of $A^{2}$ is $\alpha+\beta\sqrt2$ …
  22. Let $A$ be a matrix such that $A\cdot\begin{bmatrix}1&2\\0&3\end{bmatrix}$ is a scalar matrix and $|3A|=108$. Then $A^{2}$ equals
  23. Let $I=\begin{pmatrix}1&0\\0&1\end{pmatrix}$ and $P=\begin{pmatrix}2&0\\0&3\end{pmatrix}$. Let $Q=\begin{pmatrix}x&y\\z&4\end{pmatrix}$ for some non-zero real numbers …
  24. Let $P=\begin{bmatrix}1&0&0\\3&1&0\\9&3&1\end{bmatrix}$ and $Q=[q_{ij}]$ be two $3\times3$ matrices such that $Q-P^{5}=I_3$. Then $\frac{q_{21}+q_{31}}{q_{32}}$ is equal …
  25. If $B=\begin{bmatrix}5&2\alpha&1\\0&2&1\\\alpha&3&-1\end{bmatrix}$ is the inverse of a $3\times3$ matrix $A$, then the sum of all values of $\alpha$ for which …
  26. Let $A=\begin{bmatrix}2&3\\a&0\end{bmatrix}$, $a\in R$ be written as $P+Q$ where $P$ is a symmetric matrix and $Q$ is skew symmetric matrix. If $\det(Q)=9$, then the …
  27. If $A=\begin{bmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{bmatrix}$, then the matrix $A^{-50}$ when $\theta=\frac\pi{12}$, is equal to:
  28. Let $A=\begin{bmatrix}1&0&0\\0&4&-1\\0&12&-3\end{bmatrix}$. Then the sum of the diagonal elements of the matrix $(A+I)^{11}$ is equal to:
  29. If a point $P(\alpha,\beta,\gamma)$ satisfying …
  30. If $A$ is a symmetric matrix and $B$ is a skew-symmetric matrix such that $A+B=\begin{bmatrix}2&3\\5&-1\end{bmatrix}$, then $AB$ is equal to:
  31. Suppose $A$ is any $3\times3$ non-singular matrix and $(A-3I)(A-5I)=O$, where $I=I_3$ and $O=O_3$. If $\alpha A+\beta A^{-1}=4I$, then $\alpha+\beta$ is equal to
  32. Let $A$ be any $3\times3$ invertible matrix. Then which one of the following is not always true ?
  33. Let $J_{n,m}=\int_{0}^{1/2}\frac{x^{n}}{x^{m}-1}dx$, $\forall n>m$ and $n,m\in N$. Consider a matrix $A=[a_{ij}]_{3\times3}$ where …
  34. If …
  35. Let $\beta$ be a real number. Consider the matrix $$A=\begin{pmatrix}\beta&0&1\\2&1&-2\\3&1&-2\end{pmatrix}.$$ If $A^{7}-(\beta-1)A^{6}-\beta A^{5}$ is a singular …
  36. The number of square matrices of order $5$ with entries from the set $\{0,1\}$, such that the sum of all the elements in each row is $1$ and the sum of all the elements …
  37. Let $x\in\mathbb{R}$ and let $P=\begin{bmatrix}1&1&1\\0&2&2\\0&0&3\end{bmatrix}$, $Q=\begin{bmatrix}2&x&x\\0&4&0\\x&x&6\end{bmatrix}$ and $R=PQP^{-1}$. Then which of the …
  38. Let $M=(a_{ij}),\ i,j\in\{1,2,3\}$, be the $3\times3$ matrix such that $a_{ij}=1$ if $j+1$ is divisible by $i$, otherwise $a_{ij}=0$. Then which of the following …
  39. Let $A=\begin{bmatrix}\frac1{\sqrt{10}}&\frac3{\sqrt{10}}\\\frac{-3}{\sqrt{10}}&\frac1{\sqrt{10}}\end{bmatrix}$ and $B=\begin{bmatrix}1&-i\\0&1\end{bmatrix}$, where …
  40. If $P=\begin{bmatrix}\frac{\sqrt3}{2}&\frac12\\-\frac12&\frac{\sqrt3}{2}\end{bmatrix}$, $A=\begin{bmatrix}1&1\\0&1\end{bmatrix}$ and $Q=PAP^{T}$, then $P^{T}Q^{2015}P$ …
  41. Let $A=\begin{bmatrix}1&0&0\\1&1&0\\1&1&1\end{bmatrix}$ and $B=A^{20}$. Then the sum of the elements of the first column of $B$ is
  42. Let $$M=\begin{bmatrix}0&1&a\\1&2&3\\3&b&1\end{bmatrix}\ \text{and}\ \operatorname{adj}M=\begin{bmatrix}-1&1&-1\\8&-6&2\\-5&3&-1\end{bmatrix}$$ where $a$ and $b$ are …
  43. Let $$M=\begin{bmatrix}\sin^4\theta & -1-\sin^2\theta\\ 1+\cos^2\theta & \cos^4\theta\end{bmatrix}=\alpha I+\beta M^{-1},$$ where $\alpha=\alpha(\theta)$ and …
  44. Consider the matrix $$M=\begin{bmatrix}2&-1\\1&0\end{bmatrix}.$$ Let $p,q,r,s,a,b,c$ and $d$ be integers such that …
  45. Let $\mathbb{R}$ denote the set of all real numbers and let $i=\sqrt{-1}$. Consider the matrices $$S=\begin{bmatrix}0&-1\\1&0\end{bmatrix}\text{ and …
  46. If $A=\begin{bmatrix}5a&-b\\3&2\end{bmatrix}$ and $A\cdot\operatorname{adj}A=AA^{T}$, then $5a+b$ is equal to
  47. Let $A=\begin{bmatrix}-1&1&-1\\1&0&1\\0&0&1\end{bmatrix}$ satisfy …
  48. Let $A=\left[a_{ij}\right]$, $a_{ij}\in Z\cap[0,4]$, $1\le i,j\le2$. The number of matrices $A$ such that the sum of all entries is a prime number $p\in(2,13)$ is
  49. Let $A=\begin{bmatrix}1&-1&0\\0&1&-1\\0&0&1\end{bmatrix}$ and $B=7A^{20}-20A^{7}+2I$, where $I$ is an identity matrix of order $3\times3$. If $B=[b_{ij}]$, then $b_{13}$ …
  50. Let $A$ be a symmetric matrix such that $|A|=2$ and $\begin{bmatrix}2&1\\3&\frac32\end{bmatrix}A=\begin{bmatrix}1&2\\\alpha&\beta\end{bmatrix}$. If the sum of the …

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