Decide whether \(\begin{pmatrix}2&1\\3&4\end{pmatrix}\) is invertible using its determinant.

Decide whether \(\begin{pmatrix}1&2\\2&4\end{pmatrix}\) is invertible using its determinant.

If A is \(3\times 3\) and \(\det(A)\) = -7, what can you conclude about \(Ax=b\) for every b in \(\mathbb R^3\)?

If \(\det(A)\) = 0 for a square matrix A, what can you conclude about \(Ax=0\)?

Find all real t such that \(A=\begin{pmatrix}t&2\\8&t\end{pmatrix}\) is invertible.

A triangular \(4\times 4\) matrix has diagonal entries 1, -2, 3, and 0. Is it invertible?

If A and B are invertible square matrices, explain why \(AB\) is invertible using determinants.

Suppose \(\det(A)\) = 5 and \(\det(B)\) = 0 for square matrices of the same size. Is \(AB\) invertible?

For \(A=\begin{pmatrix}1&2&3\\0&1&4\\0&0&5\end{pmatrix}\), decide invertibility without row reducing.

A learner says a matrix with a negative determinant is not invertible. Diagnose the error.