Page 87 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

We want to reduce A to the identity matrix by row operations and simultaneously apply these operations to I to produce . To
accomplish this we shall adjoin the identity matrix to the right side of A, thereby producing a matrix of the form

Then we shall apply row operations to this matrix until the left side is reduced to I; these operations will convert the right side to
, so the final matrix will have the form

The computations are as follows:

Thus,

Often it will not be known in advance whether a given matrix is invertible. If an matrix A is not invertible, then it cannot be
reduced to by elementary row operations [part (c) of Theorem 3]. Stated another way, the reduced row-echelon form of A has
at least one row of zeros. Thus, if the procedure in the last example is attempted on a matrix that is not invertible, then at some
point in the computations a row of zeros will occur on the left side. It can then be concluded that the given matrix is not
invertible, and the computations can be stopped.

EXAMPLE 5 Showing That a Matrix Is Not Invertible
Consider the matrix