Matrix Multiplication

Matrix-Vector Product

Multiply a 3x3 matrix A by a column vector x to produce a 3-component output y. Each component y[i] is the dot product of row i of A with x, built up term by term with a running sum — the same dot-product idea extended to every row simultaneously.

Example

Use row-dot-vector products to fill the output vector.

highlighted = computed this step

Step 1 — Set up

Start with the given matrix data.

A=[21013−1024],B=[123],C=[□□□]A=\begin{bmatrix}2&1&0\\1&3&-1\\0&2&4\end{bmatrix},\quad B=\begin{bmatrix}1\\2\\3\end{bmatrix},\quad C=\begin{bmatrix}\square\\\square\\\square\end{bmatrix}

Step 2 — Top-Left entry

Fill top-left entry: 2*1 + 1*2 + 0*3 = 4.

C=[4□□],2⋅1+1⋅2+0⋅3=4C=\begin{bmatrix}\hl{4}\\\square\\\square\end{bmatrix},\quad 2\cdot 1+1\cdot 2+0\cdot 3=4

Step 3 — Bottom-Left entry

Fill bottom-left entry: 1*1 + 3*2 + -1*3 = 4.

C=[44□],1⋅1+3⋅2+−1⋅3=4C=\begin{bmatrix}4\\\hl{4}\\\square\end{bmatrix},\quad 1\cdot 1+3\cdot 2+-1\cdot 3=4

Step 4 — Third-Row First entry

Fill third-row first entry: 0*1 + 2*2 + 4*3 = 16.

C=[4416],0⋅1+2⋅2+4⋅3=16C=\begin{bmatrix}4\\4\\\hl{16}\end{bmatrix},\quad 0\cdot 1+2\cdot 2+4\cdot 3=16

Step 5 — Result

Read the completed y matrix.

C=[4416],y completeC=\begin{bmatrix}4\\4\\16\end{bmatrix},\quad \text{y complete}
matrix-vector product A · x where A is m×n and x is n×1 yields an m×1 vector y. Each entry y[i] = sum of A[i,k] * x[k] for k = 1..n.
row-dot-product view Each output entry is a dot product: row i of A times the vector x. The state panel tracks the running sum for each row, showing how the contributions accumulate.