Matrix Multiplication

Matrix Multiplication (2x3 · 3x2)

Multiply a 2x3 matrix A by a 3x2 matrix B to get a 2x2 result C. The inner dimension (3) must match. Each of the four output entries uses three terms in its dot product, and the state panel accumulates them one at a time until the cell is filled.

inner-dimension matching For A (m×p) · B (p×n), the inner dimension p must match. The output is m×n. Here A is 2×3 and B is 3×2, so p=3 and the result is 2×2.
non-square multiplication Matrix multiplication does not require square matrices. Shapes are determined by the outer dimensions after the inner dimension is consumed.

Example

Fill each 2x2 result entry from three row-column terms.

highlighted = computed this step

Step 1 — Set up

Start with the given matrix data.

A=[12340−1],B=[102−103],C=[□□□□]A=\begin{bmatrix}1&2&3\\4&0&-1\end{bmatrix},\quad B=\begin{bmatrix}1&0\\2&-1\\0&3\end{bmatrix},\quad C=\begin{bmatrix}\square&\square\\\square&\square\end{bmatrix}

Step 2 — Top-Left entry

Fill top-left entry: 1*1 + 2*2 + 3*0 = 5.

C=[5□□□],1⋅1+2⋅2+3⋅0=5C=\begin{bmatrix}\hl{5}&\square\\\square&\square\end{bmatrix},\quad 1\cdot 1+2\cdot 2+3\cdot 0=5

Step 3 — Top-Right entry

Fill top-right entry: 1*0 + 2*-1 + 3*3 = 7.

C=[57□□],1⋅0+2⋅−1+3⋅3=7C=\begin{bmatrix}5&\hl{7}\\\square&\square\end{bmatrix},\quad 1\cdot 0+2\cdot -1+3\cdot 3=7

Step 4 — Bottom-Left entry

Fill bottom-left entry: 4*1 + 0*2 + -1*0 = 4.

C=[574□],4⋅1+0⋅2+−1⋅0=4C=\begin{bmatrix}5&7\\\hl{4}&\square\end{bmatrix},\quad 4\cdot 1+0\cdot 2+-1\cdot 0=4

Step 5 — Bottom-Right entry

Fill bottom-right entry: 4*0 + 0*-1 + -1*3 = -3.

C=[574-3],4⋅0+0⋅−1+−1⋅3=−3C=\begin{bmatrix}5&7\\4&\hl{-3}\end{bmatrix},\quad 4\cdot 0+0\cdot -1+-1\cdot 3=-3

Step 6 — Result

Read the completed C matrix.

C=[574−3],C completeC=\begin{bmatrix}5&7\\4&-3\end{bmatrix},\quad \text{C complete}