# Inverse matrix

The inverse of A, written A^(-1), satisfies A times A^(-1) = I. For [[a,b],[c,d]] it is (1/(ad - bc)) times [[d,-b],[-c,a]].

Maths (university) · Part 09: Determinants, inverses and vector spaces · Card 3 of 9 · Last updated 2026-10-19

## Analogy

An undo button: the inverse reverses whatever the matrix did.

## Example

**A = [[2,1],[5,3]]**

det = 6 - 5 = 1.
Swap diagonal, negate others.
Inverse = [[3,-1],[-5,2]].
Check: row 1 dot col 1 = 6 - 5 = 1.

## In practice

Lets you solve equations by multiplying, rather than eliminating.

## Practice

### Flashcards

- **What does a zero determinant mean?** The matrix has no inverse.
- **What is the inverse of [[2,1],[5,3]]?** [[3,-1],[-5,2]].

### Interview questions

**Find the inverse of [[2,1],[5,3]] and use it to solve A x = (4, 11).**

- Determinant is 6 - 5 = 1
- Inverse is [[3,-1],[-5,2]]
- x = (3(4) - 11, -5(4) + 2(11))
- Solution (1, 2)
- Check in the original equations

## Questions people ask

### When does a matrix have an inverse?

Exactly when it is square and its determinant is not zero. Such a matrix is called invertible or non-singular, and its columns are independent. If the determinant is zero, as for [[1,2],[2,4]], the matrix squashes space and nothing can undo it.

### How do I check my inverse is correct?

Multiply the matrix by the inverse you found. You should get the identity matrix, with ones on the diagonal and zeros elsewhere. For [[2,1],[5,3]] and [[3,-1],[-5,2]], the product is [[1,0],[0,1]]. Do not skip this quick check, because one wrong sign is easy to make.

## Related concepts

- [Determinant](https://studyvisually.com/learn/science/maths-university/maths-university-part-9/maths-university-part-9-concept-1)
- [Solving with the inverse](https://studyvisually.com/learn/science/maths-university/maths-university-part-9/maths-university-part-9-concept-4)
- [Cofactor expansion](https://studyvisually.com/learn/science/maths-university/maths-university-part-9/maths-university-part-9-concept-2)
- [Rank and dimension](https://studyvisually.com/learn/science/maths-university/maths-university-part-9/maths-university-part-9-concept-8)

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