2x2 Matrix Inverse Calculator
The inverse of a 2 by 2 matrix undoes the linear transformation it represents and solves two-by-two linear systems. Enter the four entries and the calculator returns the determinant and the four entries of the inverse, or flags the matrix as singular if its determinant is zero.
Inverse formula
Matrix = | a b | | c d |
det = a*d - b*c
Inverse = (1 / det) * | d -b | | -c a |
No inverse exists when det = 0
The inverse swaps the diagonal entries, negates the off-diagonal entries, and divides every entry by the determinant. A zero determinant means the matrix is singular.
Worked example
For rows (4, 7) and (2, 6): det = 4 times 6 minus 7 times 2 = 24 minus 14 = 10. The inverse is 1 over 10 times the matrix with rows (6, minus 7) and (minus 2, 4), giving entries 0.60, minus 0.70, minus 0.20, and 0.40.
2x2 matrix inverse: frequently asked questions
How do you invert a 2 by 2 matrix?
For the matrix with rows (a, b) and (c, d), swap a and d, negate b and c, then divide every entry by the determinant ad minus bc. The inverse is 1 over the determinant times the matrix with rows (d, minus b) and (minus c, a).
When does a 2 by 2 matrix have no inverse?
When its determinant ad minus bc is zero. A zero determinant means the rows or columns are linearly dependent, so the matrix is singular and cannot be inverted. The calculator flags this case.
How can I check the inverse is correct?
Multiply the matrix by its inverse; the result should be the identity matrix with ones on the diagonal and zeros off it. Small rounding may appear in the off-diagonal entries when using decimals.
What is the determinant of a 2 by 2 matrix?
It is ad minus bc, the product of the main diagonal minus the product of the off-diagonal. It scales the inverse and tells you whether the matrix is invertible.
Sources
- NIST Digital Library of Mathematical Functions: Determinants and matrix inverses.
Reviewed by the CalculatorHub team, edited by James Graham, 19 June 2026. See our methodology.