Matrix Inverse Calculator
Linear Algebra
Intro: Paste a 2×2 or 3×3 real matrix. We’ll show det/adjugate and the resulting inverse.
Worked example
- Invert $\begin{pmatrix}2 & 5\\1 & 3\end{pmatrix}$
- Invert $\begin{pmatrix}1 & 0 & 2\\-1 & 3 & 1\\0 & 2 & 1\end{pmatrix}$
- Invert $A=\begin{pmatrix}4 & 7\\2 & 6\end{pmatrix}$
- Compute the determinant: $\det(A) = 4\cdot 6 - 7\cdot 2 = 24 - 14 = 10$.
- Compute the cofactor matrix (for 2×2: swap diagonal, negate off-diagonals): $C = \begin{pmatrix}6 & -7\\-2 & 4\end{pmatrix}$.
- Adjugate is the transpose of the cofactor matrix. For 2×2 here it’s the same shape: $\operatorname{adj}(A)=\begin{pmatrix}6 & -7\\-2 & 4\end{pmatrix}$.
- Multiply by $1/\det(A)$: $$A^{-1} = \dfrac{1}{10}\begin{pmatrix}6 & -7\\-2 & 4\end{pmatrix} = \begin{pmatrix}0.6 & -0.7\\-0.2 & 0.4\end{pmatrix}.$$
- Quick check (optional): $A\,A^{-1}=I_2$.
- Invert $\begin{pmatrix}2 & 1 & 0\\0 & 1 & 3\\1 & 0 & 2\end{pmatrix}$
FAQs
What if $det=0?$
Then the matrix is singular and not invertible.
Why choose MathGPT?
- Get clear, step-by-step solutions that explain the “why,” not just the answer.
- See the rules used at each step (power, product, quotient, chain, and more).
- Optional animated walk-throughs to make tricky ideas click faster.
- Clean LaTeX rendering for notes, homework, and study guides.
How this calculator works
- Type or paste your function (LaTeX like
\sin,\lnworks too). - Press Generate a practice question button to generate the derivative and the full reasoning.
- Review each step to understand which rule was applied and why.
- Practice with similar problems to lock in the method.