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Matrices

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:

1. 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).

2. 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.

3. 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:

1. Find all the cofactors of our matrix and put them in the matrix of cofactors.
2. Calculate the adjoint of the matrix of cofactors.

The inverse matrix will be determined using the following formula:

Calculate the inverse matrix of:

 $A\;=\;\left(\begin{matrix}-1&-4&-3\\0&4&5\\-3&-1&5\end{matrix}\right)$ A-1 =