Inverse matrix
The
inverse of a square matrix A, sometimes called a
reciprocal matrix, is a matrix A
^{-1} such that:
A·A^{-1}=I, where I is the identity matrix.
But not all nonzero n×n matrices have inverses. There is a specific condition:
A square matrix Ahas an inverse iff the determinant |A| ≠ 0.
A matrix possessing an inverse is called nonsingular, or invertible.
How to calculate the inverse matrix.
1. Method 1: Gauss-Jordan elimination.
This method can be done by augmenting the square matrix with the identity matrix of the same dimensions (A|I) and then:
- Carry out Gaussian elimination to get the matrix on the left to upper-triangular form. Check that there are no zeros on the digaonal (otherwise there is no inverse and we say A is singular).
- Working from bottom to top and right to left, use row operations to create zeros above the diagonal as well, until the matrix on the left is diagonal.
- Divide each row by a constant so that the matrix on the left becomes the identity matrix.
Once this is complete, so we have (I|A^{-1}), the matrix on the right is our inverse.
The inverse matrix is:
2. Method 2: Adjoint method.
To find the inverse matrix of a given matrix A:
- Find all the cofactors of our matrix and put them in the matrix of cofactors.
- Calculate the adjoint of the matrix of cofactors.
The inverse matrix will be determined using the following formula: