For a \(2\times 2\) matrix A, what does A being invertible imply about the equation \(Ax=b\) for every b in \(\mathbb R^2\)?

If the RREF of a \(3\times 3\) matrix A is \(I_3\), what can you conclude about A?

If a \(4\times 4\) matrix A has a zero row in echelon form, what can you conclude about invertibility?

Use the invertible matrix theorem to decide whether A is invertible if \(Ax=0\) has a non-zero solution.

A \(3\times 3\) matrix has pivot columns 1, 2, and 3. What does the invertible matrix theorem imply?

Can a \(3\times 3\) matrix be invertible if its columns do not span \(\mathbb R^3\)? Explain.

If \(A\) is an invertible \(n\times n\) matrix and \(AB=AC\), prove that \(B=C\).

For \(A=\begin{pmatrix}1&2\\2&4\end{pmatrix}\), use the homogeneous equation \(Ax=0\) to decide invertibility.

A square matrix A has RREF \(\begin{pmatrix}1&0&2\\0&1&-1\\0&0&0\end{pmatrix}\). Use the invertible matrix theorem to classify A.

A learner says a matrix is invertible if \(Ax=b\) has at least one solution for one particular b. Diagnose the error.