In Exercise, use the Gauss-Jordan method to find the inverse of the given matrix (if it...

In Exercise, use the Gauss-Jordan method to find the inverse of the given matrix (if it exists).

Reference Gauss-Jordan method

We can perform row operations on A and I simultaneously by constructing a “superaugmented matrix” [A|I ]. Theorem 3.14 shows that if A is row equivalent to I [which, by the Fundamental Theorem (d) (a), means that A is invertible], then elementary row operations will yield

(A | I ) → (I | A^–1 )

If A cannot be reduced to I, then the Fundamental Theorem guarantees us that A is

not invertible.

The procedure just described is simply Gauss-Jordan elimination performed on an n × 2n, instead of an n × (n + 1), augmented matrix. Another way to view this procedure is to look at the problem of finding A^–1 as solving the matrix equation AX=I_n for an n × n matrix X. (This is sufficient, by the Fundamental Theorem, since a right inverse of A must be a two-sided inverse.) If we denote the columns of X by x₁, . . . , x_n, then this matrix equation is equivalent to solving for the columns of X, one at a time. Since the columns of I_n are the standard unit vectors e₁, . . . , e_n, we thus have n systems of linear equations, all with coefficient matrix A:

A x ₁ = e₁, . . . , Ax_n = e_n

Since the same sequence of row operations is needed to bring A to reduced row echelon form in each case, the augmented matrices for these systems, [A | e₁], . . . , [A | e_n], can be combined as

[A | e₁ e₂ . . . e_n ] = (A | I_n )

We now apply row operations to try to reduce A to I_n, which, if successful, will simultaneously solve for the columns of A^–1, transforming In into A^–1.

We illustrate this use of Gauss-Jordan elimination with three examples.