Compute the determinant of the upper triangular matrix \(\begin{pmatrix}2&1&4\\0&3&5\\0&0&-1\end{pmatrix}\).

If two rows of a square matrix are swapped, what happens to the determinant?

If \(\det(A)=8\) and \(B\) is formed by adding \(5R_1\) to \(R_2\), find \(\det(B)\).

If \(\det(A)=-4\) and one row is multiplied by \(3\), what is the new determinant?

If \(\det(A)=6\) and \(\det(B)=\frac12\), compute \(\det(AB)\).

Verify \(\det(A^T)=\det(A)\) for \(A=\begin{pmatrix}1&2\\3&5\end{pmatrix}\).

A matrix has determinant \(10\). A row swap is followed by a row replacement. Find the final determinant.

Compute \(\det\begin{pmatrix}4&2&1\\0&-2&7\\0&0&3\end{pmatrix}\) and name the property used.

Use row replacement to compute \(\det\begin{pmatrix}1&2&0\\3&5&1\\2&4&3\end{pmatrix}\).

A determinant \(12\) is transformed by scaling one row by \(\frac12\) and then swapping two rows. Find the transformed determinant.

A triangular matrix reached after one row swap has diagonal product \(-9\). What was the original determinant?

Give a counterexample to \(\det(A+B)=\det(A)+\det(B)\).

Use determinant properties to compute \(\det\begin{pmatrix}2&4&0\\1&3&1\\0&2&5\end{pmatrix}\).

If \(A\) is \(3\times3\) and \(\det(A)=5\), compute \(\det(2A)\).

Find all \(k\) for which \(\begin{pmatrix}k&1&0\\0&k-2&4\\0&0&k+1\end{pmatrix}\) is singular.

Two row swaps and one scaling by \(-3\) produce a triangular determinant \(15\). Find the original determinant.

If \(\det(A)=0\), explain why \(AB\) is singular for every square \(B\) of the same size.

A student scales a row by \(4\), reaches triangular determinant \(28\), and reports \(\det(A)=28\). Correct the result.

Use row-scaling logic to prove that a matrix with a zero row has determinant zero.

Explain why row replacement is safe in determinant computation but row scaling must be tracked.