What augmented matrix is used to compute \(A^{-1}\) by row reduction?

In the cofactor method, what matrix is transposed before dividing by \(\det(A)\)?

Why must every row operation in \((A\mid I)\) be applied to both halves?

What conclusion should be drawn if the left side of \((A\mid I)\) cannot be reduced to \(I\)?

Compute the inverse of \(\begin{pmatrix}1&1\\0&2\end{pmatrix}\) using row reduction.

For \(A=\begin{pmatrix}2&1\\1&1\end{pmatrix}\), compute the cofactor matrix and hence \(A^{-1}\).

Use \((A\mid I)\) to find the inverse of \(A=\begin{pmatrix}1&3\\0&1\end{pmatrix}\).

Compute \(A^{-1}\) for \(A=\begin{pmatrix}3&0\\2&1\end{pmatrix}\) by row reduction.

Compute \(\begin{pmatrix}1&2\\3&7\end{pmatrix}^{-1}\) by row reduction and verify the result.

Use the cofactor formula to compute \(\begin{pmatrix}4&1\\3&1\end{pmatrix}^{-1}\).

A row-reduction attempt reaches \(\left(\begin{array}{cc|cc}1&2&1&0\\0&0&-3&1\end{array}\right)\). What does this show about the inverse?

Explain why the transpose in \(A^{-1}=\frac1{\det(A)}C^T\) cannot be skipped.

Use row reduction to find the inverse of \(A=\begin{pmatrix}1&0&0\\2&1&0\\3&4&1\end{pmatrix}\).

For \(A=\begin{pmatrix}1&2&0\\0&1&3\\0&0&1\end{pmatrix}\), compute \(A^{-1}\).

For which values of \(t\) will the row-reduction method fail for \(A=\begin{pmatrix}t&1\\4&2\end{pmatrix}\)?

A student forms \((I\mid A)\) instead of \((A\mid I)\) and row-reduces the left side. Explain why this does not compute \(A^{-1}\).

Compute the inverse of \(A=\begin{pmatrix}2&1&0\\0&1&0\\0&3&1\end{pmatrix}\) by applying row operations to both halves.

A student computes the cofactor matrix correctly as \(C=\begin{pmatrix}2&-1\\-5&3\end{pmatrix}\) and \(\det(A)=1\), then reports \(A^{-1}=C\). What check would reveal the error?

Explain why dividing by \(\det(A)\) in the cofactor formula forces a determinant check before computing an inverse.

Compare row reduction and cofactors for inverse computation. When is each method likely to be less error-prone?