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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 151-200 of 203

  1. Let $S=\left\{m\in\mathbf Z:A^{m^2}+A^m=3I-A^{-6}\right\}$, where $A=\begin{bmatrix}2&-1\\1&0\end{bmatrix}$. Then $n(S)$ is equal to ______.
  2. Let $A$ be a $2\times2$ real matrix and $I$ be the identity matrix of order $2$. If the roots of the equation $|A-xI|=0$ be $-1$ and $3$, then the sum of the diagonal …
  3. Let $A$ be a $3\times3$ matrix such that $X^{T}AX=O$ for all nonzero $3\times1$ matrices $X=\begin{bmatrix}x\\y\\z\end{bmatrix}$. If …
  4. Let $A$, other than $I$ or $-I$, be a $2\times2$ real matrix such that $A^{2}=I$, $I$ being the unit matrix. Let $\operatorname{Tr}(A)$ be the sum of diagonal elements …
  5. If $A=\begin{bmatrix}0&-1\\1&0\end{bmatrix}$, then which one of the following statements is not correct?
  6. If $A=\frac12\begin{bmatrix}1&\sqrt3\\-\sqrt3&1\end{bmatrix}$, then :
  7. Let $\alpha$ and $\beta$ be real numbers. Consider a $3\times3$ matrix $A$ such that $A^{2}=3A+\alpha I$. If $A^{4}=21A+\beta I$, then
  8. If $A$ and $B$ are two non-zero $n\times n$ matrices such that $A^{2}+B=A^{2}B$, then
  9. Let $A,B,C$ be $3\times3$ matrices such that $A$ is symmetric and $B$ and $C$ are skew-symmetric. Consider the statements (S1) $A^{13}B^{26}-B^{26}A^{13}$ is symmetric …
  10. If $P$ is a $3\times3$ real matrix such that $P^{T}=aP+(a-1)I$, where $a>1$, then
  11. $A=\begin{bmatrix}1&0&0\\2&1&0\\3&2&1\end{bmatrix}$. If $U_1,U_2,U_3$ are column matrices satisfying $AU_1=\begin{bmatrix}1\\0\\0\end{bmatrix}$ …
  12. $A=\begin{bmatrix}1&0&0\\2&1&0\\3&2&1\end{bmatrix}$. If $U_1,U_2,U_3$ are column matrices satisfying $AU_1=\begin{bmatrix}1\\0\\0\end{bmatrix}$ …
  13. $A=\begin{bmatrix}1&0&0\\2&1&0\\3&2&1\end{bmatrix}$. If $U_1,U_2,U_3$ are column matrices satisfying $AU_1=\begin{bmatrix}1\\0\\0\end{bmatrix}$ …
  14. 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 $B^TB$ is equal to …
  15. Let $A$ be a $3\times3$ matrix such that $A+A^T=O$. If $A\begin{bmatrix}1\\-1\\0\end{bmatrix}=\begin{bmatrix}3\\3\\2\end{bmatrix}$ …
  16. For some $\alpha,\beta\in\mathbf{R}$, let $A=\begin{bmatrix}\alpha&2\\1&2\end{bmatrix}$ and $B=\begin{bmatrix}1&1\\1&\beta\end{bmatrix}$ be such that …
  17. 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 $\_\_\_\_\_\_$ .
  18. If $A=\begin{bmatrix}2&3\\3&5\end{bmatrix}$, then the determinant of the matrix $\left(A^{2025}-3A^{2024}+A^{2023}\right)$ is :
  19. For the matrices $A=\begin{bmatrix}3&-4\\1&-1\end{bmatrix}$ and $B=\begin{bmatrix}-29&49\\-13&18\end{bmatrix}$, if …
  20. Let $A$, $B$ and $C$ be three $2\times2$ matrices with real entries such that $B=(I+A)^{-1}$ and $A+C=I$. If $BC=\begin{bmatrix}1&-5\\-1&2\end{bmatrix}$ and …
  21. Let $M$ and $N$ be two $3\times3$ non-singular skew-symmetric matrices such that $MN=NM$. If $P^T$ denotes the transpose of $P$, then $M^2N^2(M^TN)^{-1}(MN^{-1})^T$ is …
  22. Let $P=\begin{bmatrix}1&0&0\\4&1&0\\16&4&1\end{bmatrix}$ and $I$ be the identity matrix of order 3. If $Q=[q_{ij}]$ is a matrix such that $P^{50}-Q=I$, then …
  23. The number of $3\times3$ matrices A whose entries are either 0 or 1 and for which the system $A\begin{bmatrix}x\\y\\z\end{bmatrix}=\begin{bmatrix}1\\0\\0\end{bmatrix}$ …
  24. Let $M$ be a $3\times3$ matrix satisfying $$M\begin{bmatrix}0\\1\\0\end{bmatrix}=\begin{bmatrix}-1\\2\\3\end{bmatrix},\ …
  25. If $P$ is a $3 \times 3$ matrix such that $P^T = 2P + I$, where $P^T$ is the transpose of $P$ and $I$ is the $3 \times 3$ identity matrix, then there exists a column …
  26. Let $\omega$ be a complex cube root of unity with $\omega\neq1$ and $P=[p_{ij}]$ be a $n\times n$ matrix with $p_{ij}=\omega^{i+j}$. Then $P^2\neq0$, when $n=$
  27. For $3\times3$ matrices $M$ and $N$, which of the following statement(s) is (are) NOT correct?
  28. Let $X$ and $Y$ be two arbitrary, $3\times3$, non-zero, skew-symmetric matrices and $Z$ be an arbitrary $3\times3$, non-zero, symmetric matrix. Then which of the …
  29. How many $3\times3$ matrices $M$ with entries from $\{0,1,2\}$ are there, for which the sum of the diagonal entries of $M^TM$ is 5?
  30. Let $S$ be the set of all column matrices $\begin{bmatrix}b_1\\b_2\\b_3\end{bmatrix}$ such that $b_1,b_2,b_3\in\mathbb{R}$ and the system of equations (in real …
  31. The trace of a square matrix is defined to be the sum of its diagonal entries. If $A$ is a $2\times2$ matrix such that the trace of $A$ is 3 and the trace of $A^3$ is …
  32. Let $M$ denote the set of all real matrices of order $3\times 3$ and let $S=\{-3,-2,-1,1,2\}$. Let …
  33. Let $A=[a_{ij}]$ be a $3\times3$ matrix such that $A\begin{bmatrix}0\\1\\0\end{bmatrix}=\begin{bmatrix}0\\0\\1\end{bmatrix}$ …
  34. The number of matrices in $\mathcal{A}$ is
  35. The number of A in $T_p$ such that A is either symmetric or skew-symmetric or both, and det(A) divisible by p is
  36. The number of A in $T_p$ such that the trace of A is not divisible by p but det(A) is divisible by p is [Note : The trace of a matrix is the sum of its diagonal entries.]
  37. The number of matrices $A$ in $\mathcal{A}$ for which the system of linear equations $$A\begin{bmatrix}x\\y\\z\end{bmatrix}=\begin{bmatrix}1\\0\\0\end{bmatrix}$$ has a …
  38. The number of A in $T_p$ such that det(A) is not divisible by p is
  39. The number of matrices $A$ in $\mathcal{A}$ for which the system of linear equations $$A\begin{bmatrix}x\\y\\z\end{bmatrix}=\begin{bmatrix}1\\0\\0\end{bmatrix}$$ is …
  40. Let $A=\begin{pmatrix}1&0&0\\2&1&0\\3&2&1\end{pmatrix}$. If $u_1$ and $u_2$ are column matrices such that $Au_1=\begin{pmatrix}1\\0\\0\end{pmatrix}$ and …
  41. If $A=\begin{bmatrix}1&0&0\\1&1&0\\1&1&1\end{bmatrix}$, the sum of the entries of $A^{20}$ is $\_\_\_\_\_\_\_\_$ .
  42. Let $A=\begin{bmatrix}i&-i\\-i&i\end{bmatrix}$, $i=\sqrt{-1}$. Then, the system of linear equations …
  43. Let $P=\begin{bmatrix}-30&20&56\\90&140&112\\120&60&14\end{bmatrix}$ and $A=\begin{bmatrix}2&7&\omega^{2}\\-1&-\omega&1\\0&-\omega&-\omega+1\end{bmatrix}$ where …
  44. Let $A$ be a $3\times3$ matrix of non-negative real elements such that $A\begin{bmatrix}1\\1\\1\end{bmatrix}=3\begin{bmatrix}1\\1\\1\end{bmatrix}$. Then the maximum …
  45. Let $A$ be a $2\times2$ symmetric matrix such that $A\begin{bmatrix}1\\1\end{bmatrix}=\begin{bmatrix}3\\7\end{bmatrix}$ and the determinant of $A$ be $1$. If …
  46. Let $A=\begin{bmatrix}1&2\\0&1\end{bmatrix}$ and $B=I+\operatorname{adj}(A)+(\operatorname{adj}A)^{2}+\ldots+(\operatorname{adj}A)^{10}$. Then, the sum of all the …
  47. If $A$, $B$, and $\left(\operatorname{adj}\left(A^{-1}\right)+\operatorname{adj}\left(B^{-1}\right)\right)$ are non-singular matrices of same order, then the inverse of …
  48. Let $A=\begin{bmatrix}\dfrac{1}{\sqrt{2}}&-2\\0&1\end{bmatrix}$ and $P=\begin{bmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{bmatrix}$, $\theta>0$. If …
  49. If $A=\begin{bmatrix}1&0\\1&1\end{bmatrix}$ and $I=\begin{bmatrix}1&0\\0&1\end{bmatrix}$, then which one of the following holds for all $n\ge1$, by the principle of …
  50. Let $A$ be a square matrix all of whose entries are integers. Then which one of the following is true?

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