Enter the elements of any square matrix (2×2, 3×3, or 4×4) and get the inverse matrix calculated instantly. Select your matrix size, fill in the matrix entries, and the tool returns the inverted matrix along with the determinant. If the determinant is zero, you'll be told the matrix is singular and has no inverse.
Determinant
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Inv(1,1)
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Inv(1,2)
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Inv(1,3)
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Inv(2,1)
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Inv(2,2)
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Inv(2,3)
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Inv(3,1)
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Inv(3,2)
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Inv(3,3)
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Invertible?
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The inverse of a square matrix A is another matrix A⁻¹ such that A × A⁻¹ = I, where I is the identity matrix. Not every square matrix has an inverse — only those with a non-zero determinant are invertible.
This calculator uses Gauss-Jordan elimination. The original matrix is augmented with an identity matrix of the same size, and then row operations are applied to reduce the left side to the identity. Whatever remains on the right side is the inverse.
A determinant of zero means the matrix is singular — it has no inverse. This occurs when rows or columns are linearly dependent, meaning the matrix does not span the full dimensional space.
This calculator supports 2×2, 3×3, and 4×4 square matrices. Select your desired size from the Matrix Size dropdown and fill in the entries accordingly.
Yes. All matrix entry fields accept any real number including negatives, fractions, and decimals. For example, you can enter -3.5 or 0.25 in any cell.
Only square matrices (same number of rows and columns) can be invertible because the identity matrix I must match dimensions on both sides of the equation A × A⁻¹ = I. Non-square matrices can have pseudo-inverses but not true inverses.
Multiply your original matrix by the computed inverse. If every entry in the result equals the identity matrix (1s on the diagonal, 0s elsewhere), the inverse is correct. Small floating-point rounding errors near zero are normal.
Matrix inversion is used to solve systems of linear equations (x = A⁻¹b), compute transformations in computer graphics, perform statistical analyses like linear regression, and in control systems and engineering applications.