Invert matrix

Online calculator for inverting a 3x3 matrix

Calculate the inverse matrix

Enter the matrix to be inverted

Calculate the inverse matrix

Decimal places
M11   M12   M13
  M21   M22   M23
  M31   M32   M33

Description of the inversion of a matrix

Cramer's rule

A matrix cannot always be inverted. The following article describes this on a 2 x 2 matrix.

There is a quick way to get an inverse for a 2 x 2 matrix. This is a special case of Cramer's rule, which is used to solve systems of equations.

Die Inverse von \(\displaystyle \begin{bmatrix}a & b \\ c & d \end{bmatrix}\)   ist   \(\displaystyle \begin{bmatrix}a & b \\ c & d \end{bmatrix}^{-1} = \frac{1}{ad-bc}\begin{bmatrix}d & -b\\-c & a\end{bmatrix}\)

There are three steps to inverting a 2x2 matrix:

  • Swap the diagonal elements

  • Change the sign of the other elements

  • Divide each element by \(ad-bc \)

A matrix cannot always be inverted

Assume \(ad = bc \) in the formula above. Then we get \(ad - bc \) = 0, and we would try to divide by zero. So there is consequently no reversal. In this case the original matrix A is called a singular matrix. If the matrix has an inverse, the matrix is not singular.

Another way to get \(ad = bc \) is if the second row of the matrix is a multiple of the first.

Without actually calculating the inverse matrix, one can decide whether an inverse exists by simply calculating a single number, the denominator in the formula. This denominator is called the determinant.

If the determinant is zero, it is a singular matrix, so it cannot be inverted.

The Cramers rule also exists for larger matrices, but is computationally very inefficient. It is therefore helpful, especially for large matrices, to be able to determine whether the inverse exists before starting. This can be done by defining the determinant of the matrix for large matrices as well.

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