Matrix Inverse Calculator

Solve A⁻¹ = adj(A)/det(A) step by step — free matrix inverse calculator with worked examples, real-world defaults and instant answers.

A⁻¹ = adj(A)/det(A)

Matrix A

[ 3  -1 ]
[ -5  2 ]

A⁻¹

1
Determinant first:
det = (2)(3) − (1)(5) = 1
2
Swap the diagonal, negate the off-diagonal (the adjugate):
adj = [ 3 -1 ] [ -5 2 ]
3
Divide by the determinant:
A⁻¹ = (1/1) × adj = [ 3 -1 ] [ -5 2 ]
4
Check — A·A⁻¹ must be the identity:
[ 1 0 ] [ 0 1 ]

The inverse undoes the matrix: A⁻¹(Ax) = x, which is how matrix equations Ax = b get solved as x = A⁻¹b. The 2×2 swap-and-negate shortcut is worth memorizing; beyond that, Gauss-Jordan on [A | I] is the systematic route.

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Matrix inverse — adjugate method for 2×2, Gauss-Jordan for 3×3.

About Matrix Inverse Calculator

The Matrix Inverse Calculator solves A⁻¹ = adj(A)/det(A) for you with the full working shown — every substitution, every unit, every step, exactly the way a good teacher writes it on the board. The inverse undoes the matrix: A⁻¹(Ax) = x, which is how matrix equations Ax = b get solved as x = A⁻¹b.

How to use Matrix Inverse Calculator

1Enter your known values in the input fields (sensible real-world defaults are pre-filled).
2The tool substitutes them into A⁻¹ = adj(A)/det(A) and recomputes live.
3Read the answer in the result box, then expand the step-by-step solution to see the full working.
4Copy the method into your notebook, or change inputs to explore how the result behaves.

Why use Matrix Inverse Calculator?

✓Solves A⁻¹ = adj(A)/det(A) instantly as you type — no submit button, no waiting
✓Step-by-step solution shown for every calculation, not just the final answer
✓Realistic example values pre-loaded so you can see a worked example immediately
✓100% free, no sign-up, runs entirely in your browser — your numbers never leave your device
✓Mobile-friendly and fast enough to use mid-homework or mid-lesson

Frequently asked questions

What formula does the Matrix Inverse Calculator use?+

It uses A⁻¹ = adj(A)/det(A). The steps section shows the formula with your actual numbers substituted, so you can follow (and verify) every stage of the calculation rather than trusting a black box.

What should I keep in mind when using this calculator?+

Is the Matrix Inverse Calculator free to use?+

Completely free, with no sign-up or limits. All computation happens client-side in your browser, so it is fast, private, and works even on slow connections once the page has loaded.

Can I use this for homework, teaching or exam prep?+

That is exactly what it is built for. The step-by-step working mirrors how solutions are presented in class, so you can check homework, build worked examples for teaching, or practise method recall before exams.

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