What the determinant tells you
The determinant is a single number computed from a square matrix. Geometrically, its absolute value is the factor by which the matrix scales area (for a 2×2 matrix) or volume (for a 3×3 matrix), and its sign tells you whether orientation is reversed. Algebraically, a square matrix is invertible exactly when its determinant is not zero.
For a 2×2 matrix with rows (a, b) and (c, d) the determinant is ad − bc. For a 3×3 matrix we expand along the first row: a₁₁M₁₁ − a₁₂M₁₂ + a₁₃M₁₃, where each M is the determinant of the 2×2 matrix left after deleting a row and a column. The alternating signs follow the checkerboard pattern + − + / − + − / + − +.
How the inverse is found with the adjugate
The inverse of an invertible matrix A is A⁻¹ = adj(A) / det(A). The adjugate is the transpose of the cofactor matrix, where the cofactor of entry (i, j) is (−1)i+j times the determinant of the minor obtained by deleting row i and column j.
The calculator shows each stage: the minors, the cofactor matrix, its transpose, and finally the division by the determinant. It then multiplies A by the result and confirms that the product is the identity matrix, which is the definition of an inverse.
Singular matrices
When the determinant is zero the matrix is called singular and no inverse exists. This happens when one row (or column) is a combination of the others, so the matrix collapses space onto a smaller dimension and the transformation cannot be undone. In that case the calculator reports the zero determinant instead of dividing by it.
Exact fractions, not rounded decimals
Entries may be integers, decimals such as 2.5, or fractions such as 3/4. All arithmetic is done with exact fractions, so an inverse like 1/7 is never displayed as 0.142857. You can switch to decimals for display when you need them. For matrices larger than 3×3 use the Gauss–Jordan inverse calculator, and for rank or row reduction use the matrix rank and RREF tools.
Worked examples
These examples are computed by the same engine as the calculator above, so you can compare each step with your own working.
A 3×3 matrix with a non-zero determinant
First row of A⁻¹
13/19, -1/19, -7/19
Step-by-step solution
Formula used
A⁻¹ = adj(A) / det(A)
NoteGiven matrix
A =2130−14125
CalculateDeterminant by cofactor expansion along row 1
det(A) = 2·(−1·5 − 4·2) − 1·(0·5 − 4·1) + 3·(0·2 − (−1)·1) = 2·(−13) − 1·(−4) + 3·(1) = -19
det(A) = -19 ≠ 0, so A is invertible.
CalculateCofactor matrix C
C =−134117−37−8−2
CalculateAdjugate = transpose of C
adj(A) =−131747−81−3−2
ResultDivide by the determinant: A⁻¹ = adj(A) / det(A)
A⁻¹ =1319−119−719−419−719819−119319219
CheckVerify A · A⁻¹ = I
A·A⁻¹ =100010001
Frequently asked questions
How do you find the determinant of a 3×3 matrix?
Expand along the first row: multiply each entry by the determinant of its 2×2 minor, using alternating signs + − +, and add the three terms. The calculator shows every minor and every product.
When does a matrix have no inverse?
A square matrix has no inverse when its determinant is zero. Equivalent statements: its rows are linearly dependent, its rank is less than its size, and the system Ax = 0 has non-trivial solutions.
What is the inverse of a 2×2 matrix?
For rows (a, b) and (c, d) with ad − bc not equal to 0, the inverse is 1/(ad − bc) times the matrix with rows (d, −b) and (−c, a).
Can the determinant be negative?
Yes. A negative determinant means the transformation reverses orientation, for example a reflection. Its absolute value still gives the scaling factor of area or volume.