Define the rank of a matrix \(A\).

What is the largest possible rank of a \(3\times5\) matrix?

What is the rank of the zero \(2\times3\) matrix?

What is the rank of the identity matrix \(I_3\)?

Find the rank of \(\begin{pmatrix}1&2\\2&4\end{pmatrix}\).

Find the rank of \(\begin{pmatrix}1&0\\0&1\\0&0\end{pmatrix}\).

A row-reduced matrix has pivots in columns \(1\), \(3\), and \(4\). What is its rank?

Find the rank of \(\begin{pmatrix}1&2&3\\0&0&0\\0&1&1\end{pmatrix}\).

Row-reduce enough to find the rank of \(A=\begin{pmatrix}1&2&3\\1&3&4\\2&5&7\end{pmatrix}\).

Find the rank of \(A=\begin{pmatrix}1&0&2\\0&1&3\end{pmatrix}\).

For \(A=\begin{pmatrix}1&2&3\\0&1&1\\1&3&4\end{pmatrix}\), identify a basis for \(\operatorname{Col}(A)\).

Explain why row operations can identify pivot columns but the original columns are used for a column-space basis.

Find the rank of \(\begin{pmatrix}1&1&1\\1&2&3\\1&3&5\end{pmatrix}\).

Find the rank of the linear map whose matrix is \(\begin{pmatrix}1&0&1\\0&1&1\\1&1&2\end{pmatrix}\).

For which \(t\) does \(A_t=\begin{pmatrix}1&0&1\\0&1&1\\0&0&t\end{pmatrix}\) have rank \(3\)?

For which \(a\) does \(\begin{pmatrix}1&a\\a&1\end{pmatrix}\) have rank \(1\)?

For which \(b\) does \(\begin{pmatrix}1&0&1\\0&1&1\b&b&1\end{pmatrix}\) have full rank?

A student says rank is the number of non-zero entries in a matrix. Give a counterexample.

Prove that \(\operatorname{rank}A\le\min(m,n)\) for \(A\in\mathbb R^{m\times n}\).

Explain why rank counts independent outputs in a linear physics model \(\mathbf y=A\mathbf x\).