Matrix operations

What the problems look like

A few statements from this skill. Answers, hints and worked solutions live in the practice session.

1. Warm-up

   Two gradient matrices from a training step are $A = \begin{pmatrix} -1 & 4 & -2 \\ 1 & 3 & -3 \end{pmatrix}$ and $B = \begin{pmatrix} -4 & 3 & 0 \\ 4 & -1 & -1 \end{pmatrix}$. Compute their Frobenius inner product $\langle A, B \rangle_F = \operatorname{tr}(AB^{T}) = \sum_{i,j} A_{ij}B_{ij}$.

2. Warm-up

   Two gradient matrices from a training step are $A = \begin{pmatrix} 3 & 4 & 4 \\ 3 & 2 & -2 \end{pmatrix}$ and $B = \begin{pmatrix} -1 & -2 & 4 \\ 2 & -4 & -3 \end{pmatrix}$. Compute their Frobenius inner product $\langle A, B \rangle_F = \operatorname{tr}(AB^{T}) = \sum_{i,j} A_{ij}B_{ij}$.

3. Warm-up

   Two gradient matrices from a training step are $A = \begin{pmatrix} -2 & -4 & 0 \\ -4 & 0 & 3 \end{pmatrix}$ and $B = \begin{pmatrix} 2 & 2 & 2 \\ 3 & -2 & 1 \end{pmatrix}$. Compute their Frobenius inner product $\langle A, B \rangle_F = \operatorname{tr}(AB^{T}) = \sum_{i,j} A_{ij}B_{ij}$.

Problems shown under their original terms: Generated (template, solver-computed).