Matrix Inverse Calculator

Understanding the matrix inverse calculator — concept and principle

The inverse of a square matrix A is the matrix A⁻¹ that undoes A: multiplying them in either order gives the identity matrix, A·A⁻¹ = A⁻¹·A = I. It is used to solve Ax = b as x = A⁻¹b, to reverse transformations in graphics and robotics, and in statistics with covariance matrices. The calculator also reports det(A), which tells you at a glance whether an inverse exists.

How it is calculated

Enter the matrix row by row, separating numbers with spaces and rows with a semicolon. The calculator writes the augmented matrix [A | I] and applies Gauss-Jordan elimination: it swaps rows, scales rows and adds multiples of one row to another until the left half becomes I. The same operations applied to the right half turn I into A⁻¹, so the result is [I | A⁻¹]. If a zero pivot column appears that cannot be fixed by a row swap, the determinant is 0 and the matrix is reported as not invertible.

For a 2×2 matrix there is a shortcut: if A = [a b; c d], then det(A) = ad − bc and A⁻¹ = (1/det) · [d −b; −c a].

The principle

Every elementary row operation is the same as multiplying on the left by a simple invertible matrix E. Reducing A to I means E_k⋯E_2E_1·A = I, so the product E_k⋯E_1 must be A⁻¹. Writing I next to A simply records that product as you go. The inverse exists exactly when the columns of A are linearly independent, which is the same as det(A) ≠ 0: a matrix with det = 0 squashes space into a lower dimension, and information lost that way cannot be recovered.

Worked example

1. Default input: A = [4 7; 2 6].
2. Determinant: 4·6 − 7·2 = 24 − 14 = 10, which is not 0, so A is invertible.
3. Swap the diagonal entries and change the sign of the others: [6 −7; −2 4].
4. Divide by 10: A⁻¹ = [0.6 −0.7; −0.2 0.4].
5. Check: first row of A times first column of A⁻¹ is 4·0.6 + 7·(−0.2) = 2.4 − 1.4 = 1, and 4·(−0.7) + 7·0.4 = 0, as required for the identity.

How to read the result

det(A⁻¹) = 1/det(A), so here it is 0.1. A determinant very close to 0 relative to the size of the entries warns that the matrix is ill-conditioned: small rounding or measurement errors in A can create large errors in A⁻¹. In that case the printed inverse may be numerically unreliable even though it exists in theory.

Common mistakes

• Trying to invert a non-square matrix; only square matrices have a two-sided inverse.
• Inverting each entry separately (1/aᵢⱼ), which is not the matrix inverse.
• Assuming (AB)⁻¹ = A⁻¹B⁻¹; the correct rule reverses the order: (AB)⁻¹ = B⁻¹A⁻¹.

Frequently asked questions

What does "not invertible (det = 0)" mean?

The rows (or columns) are linearly dependent, so A maps different vectors to the same result and cannot be undone. Ax = b then has either no solution or infinitely many.

Is the inverse of a symmetric matrix symmetric?

Yes. If A = Aᵀ and A is invertible, then (A⁻¹)ᵀ = (Aᵀ)⁻¹ = A⁻¹.

Sources: Gilbert Strang, Introduction to Linear Algebra, Wellesley-Cambridge Press — chapter on inverse matrices and Gauss-Jordan elimination · Gene H. Golub and Charles F. Van Loan, Matrix Computations, Johns Hopkins University Press