Matrix fundamentals under time pressure

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Problem

Let

A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}.

Compute $\det(A)$ , $\operatorname{tr}(A)$ , $A^{-1}$ , and the Frobenius norm $\lVert A \rVert_F$ . State whether $A$ is invertible and justify the answer in one sentence.

Reveal answer or reference solution

\det(A) = -2, \qquad \operatorname{tr}(A) = 5, \qquad A^{-1} = \begin{bmatrix} -2 & 1 \\ \frac{3}{2} & -\frac{1}{2} \end{bmatrix}, \qquad \lVert A \rVert_F = \sqrt{30}.

$A$ is invertible because its determinant is nonzero.

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Appears in

Linear Algebra Practice