Determinants

The value of $\begin{vmatrix} \sin^2 \theta & \cos^2 \theta \\ -1 & 1 \end{vmatrix}$ is:

Evaluate: $\begin{vmatrix} 1 & \omega \\ \omega & \omega^2 \end{vmatrix}$ where $\omega$ is a complex cube root of unity.

If $A$ is a $3 \times 3$ matrix and $|A| = k$, then $|-2A|$ is:

Find the cofactor $A_{12}$ for the determinant $\begin{vmatrix} 2 & -3 & 5 \\ 6 & 0 & 4 \\ 1 & 5 & -7 \\ \end{vmatrix}$.

Which of the following sets of points are collinear?

The minor of the element 0 in the determinant $\begin{vmatrix} 5 & 0 \\ 1 & 2 \\ \end{vmatrix}$ is:

If $A$ is a square matrix of order 4 and $|A| = 2$, then $|\text{adj } A|$ is:

If $A$ is an identity matrix of order $n$, then $|\text{adj } A|$ is:

If $\text{adj } A = \begin{bmatrix} 2 & 3 \\ 4 & 5 \end{bmatrix}$, then $\text{adj }(A^T)$ is:

The matrix product $A \cdot A^{-1}$ results in:

Identify the element $a_{12}$ in the coefficient matrix $A$ for the system $2x + 5y = 10$ and $3x - y = 4$.

A system of equations $AX = B$ is consistent if $|A| = 2$. How many solutions does it have?