What does the identity matrix do to a vector \(\mathbf v\)?

State the defining condition for a symmetric matrix.

Compute \(D(3,-2)\) for \(D=\begin{pmatrix}4&0\\0&-1\end{pmatrix}\).

Is \(A=\begin{pmatrix}1&5\\5&2\end{pmatrix}\) symmetric?

Show that \(P=\begin{pmatrix}1&0\\0&0\end{pmatrix}\) is a projection matrix.

Find the transpose of \(A=\begin{pmatrix}1&2&3\\4&5&6\end{pmatrix}\).

What geometric action does \(R=\begin{pmatrix}0&-1\\1&0\end{pmatrix}\) perform on \((x,y)\)?

Check whether \(Q=\begin{pmatrix}0&1\\1&0\end{pmatrix}\) is orthogonal.

For \(D=\operatorname{diag}(2,-3,5)\), compute \(D(x,y,z)\) and describe the action.

Show that the zero matrix sends every vector to zero.

Verify that an orthogonal matrix preserves lengths, using \(Q^TQ=I\).

Explain why \(P^2=P\) means a projection does not change after being applied once.

For which \(a\) is \(A=\begin{pmatrix}2&a\\3&2\end{pmatrix}\) symmetric?

Find all \(k\) such that \(P=\begin{pmatrix}1&0\\0&k\end{pmatrix}\) is a projection.

For which \(c\) is \(Q=\begin{pmatrix}c&0\\0&c\end{pmatrix}\) orthogonal?

Classify \(A=\begin{pmatrix}0&1\\-1&0\end{pmatrix}\) as symmetric, diagonal, orthogonal, or projection where applicable.

A diagonal matrix \(D\) satisfies \(D(1,0)=(3,0)\) and \(D(0,1)=(0,-2)\). Find \(D\) and \(D(4,5)\).

A student says every matrix with many zeros is diagonal. Give a counterexample and explain.

Prove that the product of two diagonal \(2\times2\) matrices is diagonal.

Explain why the determinant test for invertibility cannot be applied to a \(2\times3\) zero matrix.