Matrix inversion using Cramer method

Inverse of a 2 x 2 Matrix

There is a quick method for obtaining an inverse for a 2 x 2 matrix. This is a special case of Cramer’s rule used to solve sets of equations.

The inverse of

There are three steps to invert a 2x2 matrix

• Swap the diagonal elements

• Change the sign of the other elements

• Divide each element by \(ad-bc\)

If for example in the formula above \(ad = bc\), then we would be attempting to divide by zero and consequently there is no inverse.

In this case we call the original matrix \(A\) a singular matrix. If the matrix has an inverse, we say that the matrix is non-singular.

Cramer’s rule also exists for larger matrices but is computationally very inefficient. Therefore, it is helpful especially for large matrices if we can determine before starting the calculation, whether the inverse exists. This can be done by defining also for large matrices a single number that characterizes the matrix – again, it is called the determinant of the matrix.

So we can determine whether an inverse exists by simply calculating a single number, the denominator in the formula above. This denominator is called the determinant.

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

Since Cramer's rule is computationally very inefficient for larger matrices, it will not be discussed further here. Instead, read the description of inversion using the Gauss-Jordan algorithm.

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