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Matrix Inverse Calculator

Matrix A

Square matrix of order 1 to 10. Rows and columns are synchronized. Use integers, decimals or fractions.

Matrix Inverse Result

The matrix is singular: its determinant is zero and its inverse does not exist.

How to Use the Matrix Inverse Calculator

Fill a single matrix A to find its inverse:

  1. Choose an order from 1 to 10 in Rows or Columns. Both fields stay synchronized.
  2. Confirm the size with Enter or by leaving the field, then fill each cell with an integer, decimal or fraction, such as 3/4.
  3. Read the updated inverse below the matrix or click Calculate inverse to confirm the entered size.

Empty or invalid entries are not silently converted to zero. Use zero or values with magnitude between 10⁻¹² and 10¹².

Matrix A is not replaced by its inverse. If A is singular or numerically unstable, a message explains why no reliable result is available.

Example: Inverse of a 2×2 Matrix

Consider matrix A with rows (4, 7) and (2, 6):

4
7
2
6

First, det(A) = 4 × 6 − 7 × 2 = 10. Swap 4 and 6, change the signs of 7 and 2, and divide every entry by 10.

A−1=110(6−7−24)\mathbf{A^{-1} = \frac{1}{10}\begin{pmatrix}6 & -7 \\ -2 & 4\end{pmatrix}}
3/5
-7/10
-1/5
2/5

The result has rows (3/5, −7/10) and (−1/5, 2/5). Multiplying A by this inverse gives the 2×2 identity matrix. This direct formula applies only to order 2.

What Is a Matrix Inverse?

The inverse of a matrix A is the matrix A⁻¹ that undoes its linear transformation. Its product with A, in either order, is the identity matrix.

The matrix must be square and nonsingular. In exact arithmetic, this is equivalent to det(A) ≠ 0.

Matrix Inverse Properties

  1. Identity: I⁻¹ = I.
  2. Uniqueness: when an inverse exists, it is unique.
  3. Product: (AB)⁻¹ = B⁻¹A⁻¹, if A and B are invertible and of the same order.
  4. Inverse of the inverse: (A⁻¹)⁻¹ = A.
  5. Determinant: det(A⁻¹) = 1/det(A).

The inverse helps you study linear systems and reversible transformations. Consult the determinant calculator and check the product with the matrix multiplication calculator.

Matrix Inverse Formulas and Method

For order 1, take the reciprocal of the nonzero entry. For order 2, use the formula below. The calculator uses Gauss-Jordan with row swaps to obtain [I | A⁻¹] from [A | I].

AA−1=A−1A=I\mathbf{AA^{-1} = A^{-1}A = I}
[a]−1=[1/a],a≠0\mathbf{[a]^{-1} = [1/a], \quad a \ne 0}
A−1=1ad−bc(d−b−ca),ad−bc≠0\mathbf{A^{-1} = \frac{1}{ad-bc}\begin{pmatrix}d & -b \\ -c & a\end{pmatrix}, \quad ad-bc \ne 0}
A⁻¹Inverse of matrix A.
IIdentity matrix of the same order as A.
a, b, c, dEntries of a 2×2 matrix, read row by row.

Operations are performed on a copy. Both products, A × A⁻¹ and A⁻¹ × A, are checked before rounding, with a maximum absolute error of 10⁻⁸ per entry. Nearly singular matrices may be rejected due to numerical instability.

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