List of top Mathematics Questions on Properties of Determinants

If \(A\) is a \(3 \times 3\) matrix and \( |A| = 5 \), find the value of \( |adj(A)| \).

If \(A = \begin{bmatrix} 1 & -2 & 2 0 & 2 & -3 3 & -2 & 4 \end{bmatrix}\), find the value of the matrix expression \(A(I + \text{adj } A)\), where \(I\) is the identity matrix of the same order as \(A\).

Let \( A \) be a square matrix of order \(3\times3\). If \( |A|=-4 \), then the value of \[ \left|\frac{A^{-1}}{-2}\right| \] is:

If the matrix \[ A= \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \] then the value of $|A|$ is:

If $A$ is a skew-symmetric matrix of order $3$, then $|A|$ is:

If \[ \begin{vmatrix} 1 & 1 & 1 \\ a & b & c \\ a^2 & b^2 & c^2 \end{vmatrix}=0 \] then:

If the determinant of the matrix \[ \begin{vmatrix} 1 & 2 \\ 3 & x \end{vmatrix} \] is equal to $-2$, then the value of $x$ is:

If \(A\) is a \(3 \times 3\) matrix such that \( |A| = 5 \), find the value of \( |adj(A)| \).

If determinant of a matrix is zero, then matrix is: ____.

If \(A\) is a square matrix of order \(3\) such that \( |adj\,A| = 64 \), then find the value of \( |A| \).

Let \(f(x) = \begin{vmatrix} x & 1\\ \sin(2\pi x) & 2x^2 \end{vmatrix}\). If \(f(x)\) is an odd function, \(f(-x)=g(x)\) and \(\lambda f(1)g(1)=4\), then the value of \(\lambda\) is equal to

If \(A\) is a square matrix of order \(3\) and \(|A| = 5\), find \(|adj(A)|\).

Evaluate the determinant $\begin{vmatrix} 11 & 1 & 1 \\ 1 & 21 & 1 \\ 1 & 1 & 31 \end{vmatrix}$:

If \(A\) is a \(3\times3\) matrix such that \(|A| = 4\) and \(B = \text{adj}\,A\), find the value of \(|B|\).

The vectors $\vec{p} = \hat{i} + a\hat{j} + a^2\hat{k}, \vec{q} = \hat{i} + b\hat{j} + b^2\hat{k}$ and $\vec{r} = \hat{i} + c\hat{j} + c^2\hat{k}$ are non-coplanar and $\begin{vmatrix} a & a^2 & 1+a^3 \\ b & b^2 & 1+b^3 \\ c & c^2 & 1+c^3 \end{vmatrix} = 0$ then the value of $(abc)$ is

The vectors $\vec{p} = \hat{i} + a\hat{j} + a^2\hat{k}, \vec{q} = \hat{i} + b\hat{j} + b^2\hat{k}$ and $\vec{r} = \hat{i} + c\hat{j} + c^2\hat{k}$ are non-coplanar and $\begin{vmatrix} a & a^2 & 1+a^3 \\ b & b^2 & 1+b^3 \\ c & c^2 & 1+c^3 \end{vmatrix} = 0$ then the value of $(abc)$ is

Let \( \Delta = \begin{vmatrix} x & y & 1 x+y & y+1 & x+1 1 & x & y \end{vmatrix} \). If \( x+y=-1 \), then the value of \( \Delta \) is equal to

If \(\begin{vmatrix} a & 1 & 1 1 & b & 1 1 & 1 & c \end{vmatrix}=2\), where \(a,b\) and \(c\) are positive integers, then \(a+b+c\) is equal to

Let $P=\begin{pmatrix}1 & 1 & 1 \\ 0 & 2 & 2 \\ 0 & 0 & 3\end{pmatrix}$ and $Q=\begin{pmatrix}2 & 1 & \frac{2}{3} \\ 0 & 4 & \frac{4}{3} \\ 0 & 0 & 6\end{pmatrix}$. Then $\det(QPQ^{-1})$ is equal to:

If the matrix \( \begin{pmatrix} 8-k & 2 -2 & 4-k \end{pmatrix} \) is singular, then the value of \( k \) is equal to

Let \( A \) be a \(3 \times 3\) matrix and let \( B = 3A \). If \( |A| = 5 \), then the value of \( \frac{|\text{adj } B|}{|3A|} \) is equal to

Value of the determinant of a matrix \( A \) of order \( 3 \times 3 \) is 7Then the value of the determinant formed by the cofactors of matrix \( A \) is

The cofactor of the element \( a_{21} \) in the expansion of \[ \Delta = \begin{vmatrix} 1 & 4 & 4 -3 & 5 & 9 2 & 1 & 2 \end{vmatrix} \] is

If \( A(\text{adj} A) = \begin{bmatrix} 5 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 5 \end{bmatrix} \), then the value of \( |A| + |\text{adj} A| \) is equal to :