Matrix Inverse Calculator
Compute the inverse of a 2×2 or 3×3 square matrix using its determinant and adjugate.
Inverse exists because the determinant is non-zero.
- 1
Cofactor term 1
3 × (0×1 − -2×1) = 6 - 2
Cofactor term 2
0 × (2×1 − -2×0) = 0 - 3
Cofactor term 3
2 × (2×1 − 0×0) = 4 - 4
Determinant det(A)
6 − 0 + 4 = 10Non-zero determinant confirms the matrix is invertible.
How does this calculator work?
The inverse of a square matrix A is A⁻¹ = (1/det(A))·adj(A). For a 2×2 [[a,b],[c,d]] use (1/(ad−bc))·[[d,−b],[−c,a]]; for a 3×3 use the transpose of the cofactor matrix divided by the determinant. An inverse exists only when det(A) is non-zero.
Formula
How this is calculated
Enter the square matrix A with one row per line and the entries separated by spaces or commas. The calculator detects the size n (2 or 3) from the number of rows and requires every row to have exactly n numbers; anything else is treated as invalid input.
It first computes the determinant det(A). A matrix is invertible only when det(A) ≠ 0; if the determinant is zero the matrix is singular and no inverse exists, so an error is shown. For a 2×2 matrix [[a,b],[c,d]] the inverse is (1/det)·[[d,−b],[−c,a]] where det = ad − bc. For a 3×3 matrix the inverse is (1/det)·adj(A): each cofactor C[i][j] = (−1)^(i+j)·M[i][j] uses the 2×2 minor determinant obtained by deleting row i and column j, and the adjugate is the transpose of the cofactor matrix.
The inputs are pure numbers (dimensionless), so the inverse entries carry no units. Results are rounded for display to four significant decimals but the underlying arithmetic is exact floating point. Note that near-singular matrices (determinant very close to zero) can produce large, numerically sensitive inverse entries.
Frequently asked questions
A matrix has no inverse when its determinant is exactly zero. This means the rows (or columns) are linearly dependent, so the system collapses and cannot be inverted.
Only square 2×2 and 3×3 matrices. The size is inferred from the number of rows, and each row must contain the same number of entries as there are rows.
Multiply A by the reported A⁻¹. If the inverse is correct the product equals the identity matrix (ones on the diagonal, zeros elsewhere), up to small rounding.
Also known as
TG we-Calculate Editorial Team. (2026). Matrix Inverse Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/matrix-inverse-calculator
TG we-Calculate Editorial Team. "Matrix Inverse Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/matrix-inverse-calculator.
TG we-Calculate Editorial Team, "Matrix Inverse Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/matrix-inverse-calculator
@misc{wecalculate_matrix_inverse_calculator, title = {Matrix Inverse Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/matrix-inverse-calculator}}, year = {2026}, note = {TG we-Calculate} }
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