Matrix Inverse

Inverse of a 2x2 Matrix (Adjugate Method)

Compute A⁻¹ = (1/det) adj(A) for a 2x2 matrix. The adjugate swaps the diagonal entries and negates the off-diagonal entries; dividing by the determinant gives the inverse. Verified by computing A · A⁻¹.

Example

Compute determinant, adjugate, then inverse.

highlighted = computed this step

Step 1 — Set up

Set up the given matrix data.

A=[1223]A=\begin{bmatrix}1&2\\2&3\end{bmatrix}

Step 2 — Determinant

Compute the determinant: det = -1.

det=-1det=\hl{-1}

Step 3 — Adjugate

Build the adjugate by swapping and negating entries.

adj=[3-2-21]adj=\begin{bmatrix}\hl{3}&\hl{-2}\\\hl{-2}&\hl{1}\end{bmatrix}

Step 4 — Scale by determinant

Read the inverse matrix A inverse.

Ainv=[-322-1],A inverse completeAinv=\begin{bmatrix}\hl{-3}&\hl{2}\\\hl{2}&\hl{-1}\end{bmatrix},\quad \text{A inverse complete}
adjugate formula For A = [[a,b],[c,d]], adj(A) = [[d,-b],[-c,a]]. Dividing by det(A) gives A⁻¹ = (1/det) adj(A). Requires det(A) ≠ 0.