ML Prep

Matrix fundamentals under time pressure

Problem

Let

A=[1234].A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}.

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

Reveal answer or reference solution
det⁡(A)=−2,tr⁡(A)=5,A−1=[−2132−12],∥A∥F=30.\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}.

AA is invertible because its determinant is nonzero.

Outcome

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