Matrix A
Choose size, enter values, then calculate. Use Tab to move between cells. Fractions like 1/2 are accepted.
Output
Linear system AX = B
Enter column vector B (same dimension as A). If A is invertible, X = A⁻¹B.
Practice mode
Random 2×2 with integer entries. Find the inverse, then reveal the answer.
What is an inverse matrix?
For a square matrix A, the inverse A⁻¹ (when it exists) is the unique matrix such that A·A⁻¹ = A⁻¹·A = I, where I is the identity matrix. You can use A⁻¹ to solve linear systems AX = B by multiplying on the left: X = A⁻¹B.
When does the inverse exist?
A square matrix is invertible if and only if its determinant is non-zero. If det(A) = 0, the matrix is singular and has no inverse.
Applications
- Solving linear equations in engineering and science
- Computer graphics (transformations between coordinate systems)
- Machine learning and statistics (e.g. normal equations, some optimization steps)
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FAQ
- How do I find the inverse of a matrix?
- For small matrices, use cofactors and the adjugate: A⁻¹ = (1/det(A))·adj(A). For 2×2, there is a direct formula. This calculator also uses row reduction internally for a stable numeric inverse.
- What if the determinant is zero?
- Then the matrix is not invertible. The rows (and columns) are linearly dependent; the system AX = B may have no solution or infinitely many.
- What is the adjugate matrix?
- The adjugate is the transpose of the cofactor matrix. Its entries are signed minors. For invertible A, A⁻¹ = adj(A)/det(A).
- Why is the inverse matrix important?
- It lets you "undo" a linear transformation and solve square linear systems explicitly, which shows up everywhere from circuits to 3D transforms to regression.