List of top Mathematics Questions on Matrices and Determinants

If $3A+4B^{T}=\begin{bmatrix}7&-10& 17\\ 0& 6& 31\end{bmatrix}$ and $2B-3A^{T}=\begin{bmatrix}-1& 18\\ 4&-6\\-5&-7\end{bmatrix}$ then $B=$

If A and B are 4x4 matrices such that $A^{2}+B=A^{2}B$ then which of the following is correct?

If A is a 4x4 matrix and $|2A|=64$, $B=adj A$ then $|adj B| =$

If A is a matrix of order 3x3 and $|adj(adj(adj A))|=12^{4}$, then the value of $|A^{-1} adj A| =$

For what value of $\lambda$ the system of equations $x+2y+\lambda z=0, x+2y+z=6, x+2y+3z=10$, has no solution.

If \( A=\begin{bmatrix}2 & 3 \\ 5 & -2\end{bmatrix} \), then:

If $A = \begin{bmatrix} 1 & 1 & 2 \\ 2 & 5 & 4 \\ 1 & 0 & 5 \end{bmatrix}$, then the determinant of $\left(A^{2026} - 11A^{2025} - 9A^{2023}\right)$ is equal to:}

If $P = \begin{pmatrix} 1 & \alpha & 3 \\ 1 & 3 & 3 \\ 2 & 4 & 4 \end{pmatrix}$ is the adjoint of a $3 \times 3$ matrix A and $det(A) = 4$, then $\alpha =$

Let \( A = \begin{bmatrix} 1 & 2 2 & 1 \end{bmatrix} \) and \( B = \begin{bmatrix} x & y 1 & 2 \end{bmatrix} \) be two matrices such that \( (A + B)(A - B) = A^2 - B^2 \).
If \( C = \begin{bmatrix} x & 2 1 & y \end{bmatrix} \), then trace \((C) = \)

If \[ A= \begin{bmatrix} \cos\alpha& 0 &\sin\alpha\\ 0& 1& 0 -\sin\alpha& 0 &\cos\alpha\\ \end{bmatrix} \] and \(A^2=A^T\) for one value of \(\alpha\in(0,\pi)\), then \(A^3=\):

Consider the system of linear equations (L): \[ 2x-y-z=-3,\qquad x+2y+z=4,\qquad 3x+y+kz=3 \] Let \(k\in N\) and \(1\le k\le2026\). If A={k:{ no solution}},
B={k:{ unique solution}},
C={k:{ infinite solutions} then \(n(A)+n(B)+n(C)=\)}

Let \(A\) be a \(3\times3\) matrix such that \[ \det(A)=-1. \] If \[ B^{-1}=Adj\!\left(A\,Adj(A^2)\right), \] then find \[ \det((\det A)B). \]

Let \(A\) be a \(3\times3\) matrix. If \[ A \begin{bmatrix} 001 \end{bmatrix} = \begin{bmatrix} 123 \end{bmatrix}, \quad A \begin{bmatrix} 101 \end{bmatrix} = \begin{bmatrix} 10-1 \end{bmatrix}, \quad A \begin{bmatrix} 110 \end{bmatrix} = \begin{bmatrix} 110 \end{bmatrix}, \] then the rank of \((A-I)\) is

Let \(S\) be a symmetric matrix obtained from \[ A= \begin{bmatrix} 1 & 2 & -3\\ 2 & -2 & 1\\ 3 & 1 & -1 \end{bmatrix} \] and \(T\) be a skew-symmetric matrix obtained from \[ B= \begin{bmatrix} 4 & 2 & 0\\ 1 & -1 & 3\\ 0 & 2 & -3 \end{bmatrix} \] If trace of \(S=-4\) and the non-zero elements of \(T\) are \(-1,1\), then \(S+T=\)

If \[ A= \begin{bmatrix} b+c & a & a\\ b & c+a & b\\ c & c & a+b \end{bmatrix} \] is a matrix such that trace of $A=18$ and \[ \det(A)=96, \] if $a,b,c\in \mathbb{N}$ and $ab=6$, then $ab+bc+ca=$

If \[ A= \begin{bmatrix} 3 & -3 & 4 \\ 2 & -3 & 4 \\ 0 & -1 & 1 \end{bmatrix}, \] then $A^{4}=$}

If \[ A= \begin{bmatrix} \frac12 & \frac{\sqrt3}{2}\\ -\frac{\sqrt3}{2} & \frac12 \end{bmatrix}, \] then $A^{10}=$}

If \[ A= \begin{bmatrix} 1& 7& 49\\ 2& 16& 130\\ 2& 18& 170 \end{bmatrix}, \] then $\det(A)=$

The rank of the matrix \[ \begin{bmatrix} -4 & -1 & 1 & 4 \\ -3 & 0 & 2 & 3 \\ -2 & -1 & 0 & -4 \end{bmatrix} \] is:

A die is thrown again and again until three 5's are obtained. The probability of obtaining the third 5 in the seventh throw of the die is:

On a matrix A when three elementary operations namely interchange of \( R_1 \) and \( R_2 \), \( R_2 \rightarrow R_2 - 2R_1 \), \( R_3 \rightarrow R_3 - 3R_1 \) are applied successively, A is transformed to \( \begin{bmatrix} 1 & 3 & 4 \\ 2 & 1 & 5 \\ 6 & 1 & 2 \end{bmatrix} \). Then \( \text{Tr}(A) = \)

For a \( 3 \times 3 \) non singular matrix A, if \( \text{Adj}(\text{Adj}(\text{Adj}(\text{Adj}(A)))) = |A|^n A \), then \( n = \)

If $\begin{bmatrix}x+y & 2 \\ 1& x-y\end{bmatrix}=\begin{bmatrix}4& 2 \\ 1& 2\end{bmatrix}$, then the values of $x$ and $y$ are:}

Which of the following systems has non-trivial solution?

If $A$ is a square matrix of order 3 and $|A|=5$, then the value of $|2A^{T}|$ is