4 Determinants Online Quiz Questions

True or false: Replacing one row of a matrix by itself plus a scalar multiple of another row leaves the determinant unchanged.

What term describes a square matrix whose determinant is zero?

The columns of $A=\begin{bmatrix}2&0\\1&4\end{bmatrix}$ form a parallelogram. What is its area?

True or false: For every square matrix $A$, $\det(A^T)=\det(A)$.

Which of the following conditions are equivalent to a square matrix being invertible? Select all correct choices.

A $3\times3$ matrix $A$ satisfies $\det(A)=-2$. What is $\det(2A)$?

For the system $\begin{bmatrix}1&2\\3&4\end{bmatrix}\begin{bmatrix}x_1\\x_2\end{bmatrix}=\begin{bmatrix}5\\6\end{bmatrix}$, which matrix is $A_2$ in Cramer's rule?

Which statements correctly describe the geometric meaning of a determinant? Select all correct choices.

Compute $\det(A)$ for $A=\begin{bmatrix}1&2&0\\3&-1&4\\2&0&5\end{bmatrix}$ by a cofactor expansion along the first row, and determine whether $A$ is invertible. Justify your conclusion.

If $A$ and $B$ are square matrices with $\det(A)=-3$ and $\det(B)=4$, what is $\det(AB)$?

An invertible matrix $A$ has $\det(A)=5$. What is $\det(A^{-1})$?

A square matrix has determinant $7$. If five times row 1 is added to row 2, what is the determinant of the resulting matrix?

True or false: A square matrix with a nonzero determinant is invertible.

What is the determinant of the upper-triangular matrix $\begin{bmatrix}2&1&-3\\0&-3&5\\0&0&4\end{bmatrix}$?

Find the determinant of $\begin{bmatrix}5&1\\2&3\end{bmatrix}$. Enter the numerical value.

A $3\times3$ matrix $A$ has $\det(A)=-2$. What is $\det(2A)$?

The columns of $A=\begin{bmatrix}3&1\\1&2\end{bmatrix}$ are two vectors in the plane. What is the area of the parallelogram they form? Enter the numerical value.

Which expression correctly relates the cofactor $C_{23}$ to its minor $M_{23}$?