What does it mean for a square matrix \(A\) to have inverse \(A^{-1}\)?

Can a non-square matrix have an inverse in this course's matrix sense?

Decide whether \(\begin{pmatrix}2&1\\5&3\end{pmatrix}\) is invertible.

Decide whether \(\begin{pmatrix}1&2\\3&6\end{pmatrix}\) is singular.

Find the inverse of \(A=\begin{pmatrix}3&1\\2&1\end{pmatrix}\).

Use an inverse to solve \(\begin{pmatrix}1&2\\0&1\end{pmatrix}\mathbf{x}=\begin{pmatrix}5\\3\end{pmatrix}\), given \(A^{-1}=\begin{pmatrix}1&-2\\0&1\end{pmatrix}\).

Check that \(B=\begin{pmatrix}3&-1\\-5&2\end{pmatrix}\) is the inverse of \(A=\begin{pmatrix}2&1\\5&3\end{pmatrix}\).

Find \(A^{-1}\) for \(A=\begin{pmatrix}4&7\\2&5\end{pmatrix}\).

Solve \(\begin{pmatrix}2&1\\1&1\end{pmatrix}\mathbf{x}=\begin{pmatrix}7\\4\end{pmatrix}\) using an inverse.

For \(A=\begin{pmatrix}a&1\\4&2\end{pmatrix}\), find the values of \(a\) for which \(A^{-1}\) exists.

A student writes \(\begin{pmatrix}2&4\\1&3\end{pmatrix}^{-1}=\begin{pmatrix}\frac12&\frac14\\1&\frac13\end{pmatrix}\). Explain the error and find the correct inverse.

Explain why \(A\mathbf{x}=\mathbf b\) has solution \(\mathbf{x}=A^{-1}\mathbf b\) when \(A\) is invertible.

For \(A(t)=\begin{pmatrix}t&2\\8&t\end{pmatrix}\), classify the values of \(t\) for which \(A\) is invertible.

Find \(a\) so that \(\begin{pmatrix}1&a\\a&9\end{pmatrix}\) has no inverse.

A square matrix has inverse. Explain why its columns cannot be proportional in the \(2\times2\) case.

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

For \(A=\begin{pmatrix}1&2\\3&5\end{pmatrix}\), compute \(A^{-1}\) and verify one multiplication order.

A student says that if \(AB=I\), then it is unnecessary to care about \(BA\). Discuss this for square matrices in this course.

Explain why a singular matrix cannot be used to solve every equation \(A\mathbf{x}=\mathbf b\) by multiplying by \(A^{-1}\).

Prove that if \(A\) is invertible, then \(A\mathbf{x}=\mathbf0\) has only the trivial solution.