Matrix Multiplication

Matrix Power

Compute A³ for a 2x2 matrix A via two sequential multiplications: first A² = A · A, then A³ = A² · A. Each step reuses the shared matmul emitter, so the state panel fills in P = A² entry by entry, then C = A³ entry by entry, with running sums shown throughout.

Example

Compute A squared first, then use it to fill A cubed.

highlighted = computed this step

Step 1 — Set up

Start with the given matrix data.

A=[1101],P=[□□□□],Q=[□□□□]A=\begin{bmatrix}1&1\\0&1\end{bmatrix},\quad P=\begin{bmatrix}\square&\square\\\square&\square\end{bmatrix},\quad Q=\begin{bmatrix}\square&\square\\\square&\square\end{bmatrix}

Step 2 — A^2 Top-Left entry

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

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

Step 3 — A^2 Top-Right entry

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

P=[12□□],1⋅1+1⋅1=2P=\begin{bmatrix}1&\hl{2}\\\square&\square\end{bmatrix},\quad 1\cdot 1+1\cdot 1=2

Step 4 — A^2 Bottom-Left entry

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

P=[120□],0⋅1+1⋅0=0P=\begin{bmatrix}1&2\\\hl{0}&\square\end{bmatrix},\quad 0\cdot 1+1\cdot 0=0

Step 5 — A^2 Bottom-Right entry

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

P=[1201],0⋅1+1⋅1=1P=\begin{bmatrix}1&2\\0&\hl{1}\end{bmatrix},\quad 0\cdot 1+1\cdot 1=1

Step 6 — A^3 Top-Left entry

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

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

Step 7 — A^3 Top-Right entry

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

Q=[13□□],1⋅1+2⋅1=3Q=\begin{bmatrix}1&\hl{3}\\\square&\square\end{bmatrix},\quad 1\cdot 1+2\cdot 1=3

Step 8 — A^3 Bottom-Left entry

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

Q=[130□],0⋅1+1⋅0=0Q=\begin{bmatrix}1&3\\\hl{0}&\square\end{bmatrix},\quad 0\cdot 1+1\cdot 0=0

Step 9 — A^3 Bottom-Right entry

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

Q=[1301],0⋅1+1⋅1=1Q=\begin{bmatrix}1&3\\0&\hl{1}\end{bmatrix},\quad 0\cdot 1+1\cdot 1=1
matrix power A^n means A multiplied by itself n times. For a square matrix, each step is a standard matrix multiply using the result of the previous step.
repeated multiplication A³ = (A · A) · A. We compute the intermediate A² first, store it, then multiply A² by A to get A³. The same row·col dot-product procedure applies at each step.