State what it means for a square matrix A to have inverse \(A^{-1}\).

Verify that \(B=\begin{pmatrix}1&0\\0&1\end{pmatrix}\) is the inverse of \(A=\begin{pmatrix}1&0\\0&1\end{pmatrix}\).

Use the \(2\times 2\) inverse formula to find the inverse of \(A=\begin{pmatrix}2&0\\0&3\end{pmatrix}\).

Find the inverse of \(A=\begin{pmatrix}1&2\\3&4\end{pmatrix}\).

Decide whether \(A=\begin{pmatrix}2&4\\1&2\end{pmatrix}\) is invertible.

Verify that \(\begin{pmatrix}2&1\\1&1\end{pmatrix}\) and \(\begin{pmatrix}1&-1\\-1&2\end{pmatrix}\) are inverses.

Solve \(Ax=b\) using \(A^{-1}\), where \(A^{-1}\) = \(\begin{pmatrix}1&-1\\-1&2\end{pmatrix}\) and b = \(\begin{pmatrix}3\\1\end{pmatrix}\).

Find all real t such that \(A=\begin{pmatrix}t&1\\4&2\end{pmatrix}\) is not invertible.

A student says every non-zero matrix has an inverse. Disprove this using \(A=\begin{pmatrix}1&2\\2&4\end{pmatrix}\).

If A and B are invertible square matrices of the same size, explain why (\(AB\))^{-1} = \(B^{-1}\)\(A^{-1}\) rather than \(A^{-1}\)\(B^{-1}\).