Matrix Calculator
Compute the determinant, inverse, or transpose of a 2×2 or 3×3 matrix with the expansion shown in full detail.
Matrix Calculator — input your values
What does this math calculator do?
The matrix calculator works with 2×2 and 3×3 matrices and performs three operations: the determinant, the inverse, and the transpose. The determinant for a 3×3 matrix is computed with the rule of Sarrus, and the inverse is built from the adjugate (transposed cofactor matrix) divided by the determinant.
Every result is fully worked: the diagonal products are shown for the determinant, the adjugate is displayed as its own matrix, and the final inverse is presented both as a formula (1/det × adjugate) and as a ready-to-read table. If the determinant is zero, the calculator reports that the matrix is singular and has no inverse.
Formula and variables
A square matrix has an inverse exactly when its determinant is non-zero. The transpose mirrors every entry across the main diagonal.
| Symbol | Meaning |
|---|---|
| a ... i | Matrix entries (row-major order) |
| det(A) | The determinant of A |
| adj(A) | The adjugate of A (transpose of the cofactor matrix) |
Worked example
Find the determinant and inverse of the 2×2 matrix [[4, 7], [2, 6]].
- det = ad − bc = 4 × 6 − 7 × 2 = 24 − 14 = 10.
- Adj = [[6, −7], [−2, 4]].
- Inverse = 1/10 × [[6, −7], [−2, 4]] = [[0.6, −0.7], [−0.2, 0.4]].
Answer: det = 10 and the inverse is [[0.6, −0.7], [−0.2, 0.4]].
Tips and common mistakes
Tips
- Check your determinant with the formula det = ad − bc for 2×2 matrices — it is easy to test by hand.
- A zero determinant means the matrix is singular: its rows are linearly dependent and no inverse exists.
- For a 3×3 determinant, Sarrus's rule only works for 3×3 matrices — for larger ones you need cofactor expansion.
- Verify an inverse by multiplying it by the original matrix: the product should be the identity matrix.
Common mistakes to avoid
- Forgetting the alternating signs (+ − + / − + − / + − +) when building the cofactor matrix.
- Taking the inverse of a singular matrix instead of recognizing the determinant is zero.
- Confusing the adjugate (transpose of cofactors) with the cofactor matrix itself.
Matrix Calculator — frequently asked questions
What is a determinant?
A single number computed from a square matrix that tells you whether it is invertible. If the determinant is zero, the matrix has no inverse; otherwise it does.
How is the 3×3 determinant computed?
With Sarrus's rule: copy the first two columns beside the matrix, multiply along the three main diagonals, subtract the three secondary diagonal products.
What is the adjugate matrix?
The transpose of the cofactor matrix. The inverse is then 1/det times the adjugate, which is the formula this calculator uses.
What does it mean for a matrix to be singular?
Its determinant is zero, so it has no inverse. Geometrically, its rows (or columns) are linearly dependent — the transformation collapses a dimension.
What is the transpose used for?
It mirrors the matrix across its main diagonal, swapping rows and columns. It appears throughout linear algebra, from dot products to machine learning weight updates.
Can I check the inverse is correct?
Multiply the inverse by the original matrix: A⁻¹A should equal the identity matrix (ones on the diagonal, zeros elsewhere).