Compute \(\det\begin{pmatrix}4&1\\2&3\end{pmatrix}\).

Can the determinant of a \(2\times3\) matrix be computed? Explain briefly.

Find \(\det\begin{pmatrix}-2&5\\3&1\end{pmatrix}\).

For first-row cofactor expansion of a \(3\times3\) matrix, what signs multiply the three first-row minors?

Compute \(\det\begin{pmatrix}6&-1\\4&2\end{pmatrix}\) and state whether the corresponding square system has a unique solution.

Compute \(\det\begin{pmatrix}1&2&0\\0&3&4\\0&0&5\end{pmatrix}\).

Expand along the first row to compute \(\det\begin{pmatrix}2&0&1\\3&4&5\\1&-2&0\end{pmatrix}\).

Compute \(\det\begin{pmatrix}0&2&1\\1&3&0\\4&-1&2\end{pmatrix}\) by first-row expansion.

Compute \(\det\begin{pmatrix}1&2&3\\0&-1&4\\2&1&0\end{pmatrix}\) by expanding along the first row.

For \(A=\begin{pmatrix}2&1&0\\1&3&2\\0&4&1\end{pmatrix}\), compute \(\det(A)\) and decide whether \(A\mathbf{x}=\mathbf b\) has a unique solution for every \(\mathbf b\in\mathbb R^3\).

Show by determinant calculation that the columns of \(\begin{pmatrix}1&2\\3&6\end{pmatrix}\) cannot give a unique solution matrix.

A student computes \(\det\begin{pmatrix}2&5\\7&1\end{pmatrix}=2\cdot5-7\cdot1\). Correct the computation.

Find all values of \(k\) such that \(\det\begin{pmatrix}k&2\\8&k\end{pmatrix}=0\).

For \(A(k)=\begin{pmatrix}1&0&k\\2&1&3\\0&4&1\end{pmatrix}\), find \(k\) such that \(\det(A(k))=0\).

Classify when \(A(a)=\begin{pmatrix}1&a\\a&1\end{pmatrix}\) gives a unique solution for every right-hand side.

For \(A(t)=\begin{pmatrix}t&1&0\\0&t&1\\0&0&t-2\end{pmatrix}\), find the parameter values for which \(\det(A(t))\ne0\).

A \(3\times3\) determinant is expanded along a row containing two zeros. Explain why this is usually a good strategy, and compute \(\det\begin{pmatrix}0&0&5\\1&2&3\\4&0&-1\end{pmatrix}\).

Prove from the \(2\times2\) determinant formula that two proportional columns give determinant zero.

A student says \(\det(A)=0\) means \(A\mathbf{x}=\mathbf b\) has no solution. Explain the precise conclusion instead.

Explain why checking that a matrix is square is not a cosmetic step before computing a determinant.