Inverse Matrix Calculator and Determinant

How the Determinant and Inverse Matrix Calculator works

The inverse of a matrix A is the matrix A⁻¹ that gives the identity when multiplied by A, the matrix version of 1 ÷ x. It only exists when the determinant is not zero. The determinant is a single number that measures how much the matrix stretches area or volume.

How the inverse matrix calculator works

For a 2 × 2 matrix [a b; c d] the formula is det = ad − bc and A⁻¹ = (1 ÷ det) × [d −b; −c a]. For 3 × 3 the determinant expands along the first row using 2 × 2 minors. Larger matrices are row-reduced: the calculator writes [A | I] and applies Gauss-Jordan elimination until the left half is I.

First example. For [1 2; 3 4], det = 1 × 4 − 2 × 3 = −2, so the inverse is (1 ÷ −2) × [4 −2; −3 1] = [−2 1; 3/2 −1/2].

Second example. [1 2; 2 4] has det = 4 − 4 = 0. The second row is twice the first, so the matrix is singular and has no inverse.

How many 2 × 2 minors does a 3 × 3 determinant need? Three, one for each entry of the row you expand along.

Determinants and inverses of common 2 × 2 matrices

Tips

• Zero determinant, no inverse. If one row is a multiple or a sum of others, det is 0.
• Check by multiplying. A × A⁻¹ should give 1s on the diagonal and 0s elsewhere.
• Solve systems with it. If Ax = b, then x = A⁻¹b, though elimination is quicker by hand.

Related tools