How do you find the inverse of a 2x2 matrix?

For a 2x2 matrix [a, b; c, d], the determinant is det = ad minus bc. The inverse is (1/det) times [d, -b; -c, a]. The inverse exists only when det is not zero; if det equals zero the matrix is singular and has no inverse.

How do you find the inverse of a 3x3 matrix?

The inverse of a 3x3 matrix is found by computing the matrix of cofactors, transposing it to form the adjugate matrix, and dividing every element by the determinant. The determinant of a 3x3 matrix is computed by cofactor expansion along the first row. If the determinant is zero, the matrix is singular and has no inverse.

When does a matrix not have an inverse?

A square matrix has no inverse when its determinant is zero; it is then called a singular matrix. Geometrically this means the matrix collapses space along at least one dimension, making the transformation irreversible. Common causes include a row or column of zeros, two identical rows or columns, or one row being a linear combination of others.

Matrix Inverse Calculator

The inverse of a matrix A is a matrix A-inverse such that A times A-inverse equals the identity matrix I. For a 2 by 2 matrix this means when you multiply the two matrices together you get a matrix with ones on the diagonal and zeros everywhere else. Finding a matrix inverse is fundamental in linear algebra and appears throughout applied mathematics: solving systems of simultaneous equations, computing least-squares regression coefficients, performing linear transformations in computer graphics, and inverting covariance matrices in statistics. Not every square matrix has an inverse. A matrix is invertible only if its determinant is non-zero. When the determinant is zero the matrix is called singular, meaning it collapses space along at least one dimension and the transformation cannot be reversed. For a 2 by 2 matrix the formula for the inverse is straightforward: swap the diagonal elements, negate the off-diagonal elements, and divide through by the determinant. For a 3 by 3 matrix the method uses cofactors and the adjugate matrix. This calculator supports both 2 by 2 and 3 by 3 matrices. Enter the matrix elements, choose the size, and the calculator returns the determinant, the full inverse matrix to four decimal places, and a numerical verification that multiplying the original matrix by its inverse gives the identity. If the determinant is zero it reports that the matrix is singular.

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Matrix size

Matrix A entries (row by row)

10.00

determinant

InvertibleYes

Trace10.00

det = 0 means the matrix is singular (no inverse exists). Result shown to 4 decimal places.

Inverse matrix A^-1:

How it works

For a 2×2 matrix [a, b; c, d]: det = ad - bc. If det ≠ 0, the inverse is (1/det) × [d, -b; -c, a].
For a 3×3 matrix, the determinant is computed by cofactor expansion along the first row. The cofactor matrix replaces each element with the signed determinant of its 2×2 minor. The adjugate is the transpose of the cofactor matrix. The inverse is adjugate divided by det. Verification multiplies A by A^-1 and checks the result is close to the identity matrix.

Worked example

2×2 matrix A = [4, 7; 2, 6]. Determinant = (4×6) - (7×2) = 24 - 14 = 10. Inverse = (1/10) × [6, -7; -2, 4] = [0.6, -0.7; -0.2, 0.4]. Verify: [4,7;2,6] × [0.6,-0.7;-0.2,0.4] = [4×0.6+7×(-0.2), 4×(-0.7)+7×0.4; ...] = [1,0;0,1]. These match the defaults pre-filled above.

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