❖ In this lesson, we have explained how to solve a linear system using the inverse of a matrix, which is a precalculus lesson tutorial. This lesson builds on our previous tutorials where we demonstrated how to find the inverse of \(2 \times 2, 3 \times 3,\) and \(4 \times 4\) matrices.

❖ To find the inverse of a matrix, you need to create an augmented matrix containing the original matrix and the identity matrix (which has the same size as the original matrix).

Then,

You need to transform the original matrix into the identity matrix using Elementary Row Operations

(means apply Reduced Row Echelon Form (RREF) to the left side of the augmented matrix).

By applying the same Elementary Row Operations to the augmented matrix for the right side, the identity matrix

(which is on the right side of the augmented matrix) will transform into the inverse of the original matrix.

So,

The identity matrix (which is on the left side of the augmented matrix) will convert into the inverse of the original matrix as long as you apply the same Elementary Row Operations for the augmented matrix.

❖ To confirm your answer, multiply the original matrix by its inverse. If the product is the identity matrix, then you have found the correct inverse.

This can be represented as:

(The matrix \(A\)) multiplied by (the inverse of the matrix \(A\)) equal to (the identity matrix) of the same size as \(A\),

so, if the inverse of matrix \(A\) exists then

\(A A^{-1} =\) Identity matrix.

where \(A\) is the original matrix, and \(A^{-1}\) is its inverse.

❖ In this lesson, we also explain when the inverse of a matrix does not exist (DNE).

Note the following cases:

If \(A\) is not a Square Matrix, then the inverse of matrix \(A\) is DNE.

If the linear system \(Ax=b\) has no solution, then the inverse of matrix \(A\) is DNE.

If the linear system \(Ax=b\) has infinitely many solutions, then the inverse of matrix \(A\) is DNE.

The Square Matrix is a matrix with

the number of rows \(=\) the number of columns.

The number of rows and columns that a matrix has is called its size, order, or dimension.