Matrix Inverse

Solve a System Using the Matrix Inverse

Given Ax = b and A⁻¹, compute x = A⁻¹b via matrix-vector multiplication. Each component of x is a dot product of a row of A⁻¹ with b. Verified by substituting x back into Ax = b.

solution via inverse If A is invertible, Ax = b has unique solution x = A⁻¹b. Computing each component requires a row dot product: x[i] = Ainv[i,·] · b.

Example

Multiply A inverse by b and verify.

highlighted = computed this step

Step 1 — Set up inverse and b

Set up A inverse and b for x = A inverse b.

Ainv=[7−2−31],b=[310],x=[□□]Ainv=\begin{bmatrix}7&-2\\-3&1\end{bmatrix},\quad b=\begin{bmatrix}3\\10\end{bmatrix},\quad x=\begin{bmatrix}\square\\\square\end{bmatrix}

Step 2 — Solve x1

Show the next matrix state.

x=[1□]x=\begin{bmatrix}\hl{1}\\\square\end{bmatrix}

Step 3 — Solve x2

Show the next matrix state.

x=[11]x=\begin{bmatrix}1\\\hl{1}\end{bmatrix}

Step 4 — Verify solution

Verify the solution vector satisfies Ax = b.

x=[11],solution verifiedx=\begin{bmatrix}1\\1\end{bmatrix},\quad \text{solution verified}