Matrix Inverse via the Adjugate Method

What is a matrix inverse?

A square matrix $A$ is invertible if there exists a matrix $A^{-1}$ with $A A^{-1} = A^{-1} A = I$

Not every square matrix has an inverse. $A$ is invertible if and only if $\det A \ne 0$. A matrix without an inverse is called singular.

The 2×2 formula

For $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ with $\det A = ad - bc \ne 0$: $A^{-1} = \dfrac{1}{\det A} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$

Swap the diagonal, negate the off-diagonal, divide by the determinant.

Step-by-step

$A = \begin{pmatrix} 2 & 1 \\ 5 & 3 \end{pmatrix}$

Step 1 — Compute $\det A$: $\det A = (2)(3) - (1)(5) = 6 - 5 = 1$

Step 2 — Build the adjugate (for $2 \times 2$): swap diagonal, negate off-diagonal: $\mathrm{adj}(A) = \begin{pmatrix} 3 & -1 \\ -5 & 2 \end{pmatrix}$

Step 3 — Divide by determinant: $A^{-1} = \dfrac{1}{1} \begin{pmatrix} 3 & -1 \\ -5 & 2 \end{pmatrix} = \begin{pmatrix} 3 & -1 \\ -5 & 2 \end{pmatrix}$

Verification

$A A^{-1} = \begin{pmatrix} 2 & 1 \\ 5 & 3 \end{pmatrix} \begin{pmatrix} 3 & -1 \\ -5 & 2 \end{pmatrix}$

• Row 1, col 1: $2(3) + 1(-5) = 6 - 5 = 1$ ✓
• Row 1, col 2: $2(-1) + 1(2) = -2 + 2 = 0$ ✓
• Row 2, col 1: $5(3) + 3(-5) = 15 - 15 = 0$ ✓
• Row 2, col 2: $5(-1) + 3(2) = -5 + 6 = 1$ ✓

$A A^{-1} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = I$ ✓

Generalization: adjugate method for any $n$

For a general $n \times n$ invertible $A$: $A^{-1} = \dfrac{1}{\det A} \, \mathrm{adj}(A)$

where $\mathrm{adj}(A)$ is the adjugate (or classical adjoint) — the transpose of the cofactor matrix. The $(i, j)$ entry of the cofactor matrix is $(-1)^{i+j} M_{ij}$ where $M_{ij}$ is the $(n-1) \times (n-1)$ determinant obtained by deleting row $i$ and column $j$.

For $n \ge 3$, this is often slower than using Gauss–Jordan elimination on $[A \mid I]$, but the adjugate formula gives a clean closed-form.

Why the inverse matters

• Solving $A \mathbf{x} = \mathbf{b}$: if $A$ is invertible, $\mathbf{x} = A^{-1} \mathbf{b}$.
• Change of basis: converting coordinates between different bases uses inverses.
• Undoing linear transformations: rotation, scaling, shear — each has an inverse that undoes it.

Common mistakes

• Forgetting to check $\det A \ne 0$. If the determinant is zero, $A^{-1}$ does not exist.
• Not negating the off-diagonal. It's $(-b, -c)$, not $(b, c)$.
• Dropping the $1/\det A$ factor. Without it you compute $\mathrm{adj}(A)$, not $A^{-1}$.
• Trying to invert a non-square matrix. Only square matrices can have a (two-sided) inverse; non-square matrices have pseudo-inverses instead.

Try it in the visualization

Modify entries of $A$ with sliders; the determinant and $A^{-1}$ update live. When $\det A = 0$, the inverse panel turns red and shows "singular — no inverse exists." The product $A A^{-1}$ also updates to confirm $I$.