Matrix Inverse Calculator (2x2)
2x2 determinant and inverse entries.
What it does
Two by two inverses solve paired linear equations in one step. Determinant ad minus bc decides everything: nonzero opens the inverse, zero closes it. The adjugate swaps the diagonal, flips the off diagonal signs, and scales by one over determinant.
The tool above runs the recipe. With [[4, 7], [2, 6]], determinant is 10 with inverse [[0.6, -0.7], [-0.2, 0.4]] and reciprocal 0.1. With [[1, 2], [3, 4]], determinant is -2 with inverse [[-2, 1], [1.5, -0.5]]. Multiplying either matrix by its inverse returns identity.
Signs decide correctness. The most common slip negates the wrong pair or forgets the reciprocal entirely. Checking A times inverse against [[1, 0], [0, 1]] catches both in seconds.
This page fixes 2x2 with finite entries. It does not do larger matrices, complex entries, or equation setup. For 3x3 and up, use elimination.
How to use
- Enter the four entries a, b, c, d.
- Read the determinant first.
- Read the inverse brackets.
- Read 1/det for scaling checks.
- Multiply back to identity to verify.
- Stop at zero determinant errors.
Formula
- a, b
- Top row entries a (left) and b (right).
- c, d
- Bottom row entries c (left) and d (right).
- 1/det
- Reciprocal multiplier of the adjugate.
Square 2x2 matrices with finite entries. At zero determinant the rows are dependent and the error names it. Scaling all entries scales the determinant quadratically.
Worked example
Example: inverse of [[4, 7], [2, 6]]
Defaults: [[4, 7], [2, 6]] with determinant 10. Zero determinant has no inverse.
- Inputs: [[4, 7], [2, 6]].
- Determinant: 4 times 6 minus 7 times 2 is 10.
- Reciprocal: 1 / 10 is 0.1.
- Inverse: 0.1 times [[6, -7], [-2, 4]] is [[0.6, -0.7], [-0.2, 0.4]].
- Multiply back gives identity. Tool shows the same three outputs.
Reference table
| Matrix | Det | Inverse |
|---|---|---|
| [[4,7],[2,6]] | 10 | [[0.6, -0.7], [-0.2, 0.4]] |
| [[1,2],[3,4]] | -2 | [[-2, 1], [1.5, -0.5]] |
| [[2,0],[0,5]] | 10 | [[0.5, 0], [0, 0.2]] |
| [[0,1],[1,0]] | -1 | [[0, 1], [1, 0]] |
| [[3,0],[0,3]] | 9 | [[0.3333, 0], [0, 0.3333]] |
Limits
- 2x2 only. Larger needs elimination.
- Finite entries. Huge values overflow.
- Exact to floating point.
- No equation setup.
Common mistakes
- Negating the diagonal instead of off diagonal.
- Skipping the 1/det scale.
- Reading rows as columns.
- Trusting a zero determinant inverse.
- Rounding before multiplying back.
Try it yourself
Quick checks. Answers sit tucked under each question.
Matrix [[1, 2], [3, 4]]?
Determinant is 4 minus 6 is -2. Inverse is [[-2, 1], [1.5, -0.5]] with -0.5 reciprocal.
Diagonal matrix [[2, 0], [0, 5]]?
Determinant is 10 minus 0 is 10. Inverse is [[0.5, 0], [0, 0.2]] with 0.1 reciprocal.
Questions
How is the determinant figured?
ad minus bc. Here 10.
What is the inverse formula?
1/det times [[d, -b], [-c, a]]. Here [[0.6, -0.7], [-0.2, 0.4]].
What happens at zero determinant?
No inverse exists. Rows are dependent, so the error explains it.
How do I check the answer?
Multiply A by its inverse to get identity [[1,0],[0,1]].
Do negative determinants work?
Yes for any nonzero determinant, including negatives. Here [[1,2],[3,4]] gives -2.
Does this do 3x3?
No. This page fixes 2x2. Larger matrices need elimination.