Question Type 4: Finding the determinant for a 2x2 matrix Exercises

The vectors $\mathbf u=\begin{pmatrix}2\\3\end{pmatrix}$ and $\mathbf v=\begin{pmatrix}1\\5\end{pmatrix}$ form a parallelogram. Calculate its area.

Find the determinant of the matrix $A = \begin{pmatrix}3 & 5 \\ 2 & 7\end{pmatrix}$.

Find the determinant of $B = \begin{pmatrix}-4 & 6 \\ 3 & -2\end{pmatrix}$.

Determine all values of $k$ for which $F=\begin{pmatrix}k+1 & 4 \\ 2 & k-3\end{pmatrix}$ is invertible.

Find the determinant of $C$.

Let $A=\begin{pmatrix}2 & 1\\0 & 3\end{pmatrix}$ and $B=\begin{pmatrix}4 & 5\\1 & 2\end{pmatrix}$. Compute $\det(A)$, $\det(B)$ and $\det(AB)$.

Compute the determinant of $H=\begin{pmatrix}t^2 & t\\2 & 3t\end{pmatrix}$ as a function of $t$.

Compute the determinant of $D = \begin{pmatrix}a & b \\ c & d\end{pmatrix}$ in terms of $a,b,c,d$.

If $A=\begin{pmatrix}1 & 2\\3 & 4\end{pmatrix}$, find $\det(3A)$.

Find the determinant of $G=\begin{pmatrix}\cos\theta & -\sin\theta\\\sin\theta & \cos\theta\end{pmatrix}$.

Given that $\det\begin{pmatrix}2 & x \\ 3 & 5\end{pmatrix}=7$, find $x$.

For the matrix $E=\begin{pmatrix}k & 2 \\ 5 & k-1\end{pmatrix}$, express $\det(E)$ in terms of $k$.