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Example 6

Find the determinantof

Remember from that at the end of the forward elimination steps of Gaussianelimination with partial pivoting, we obtained

Since rows were switched once during the forward elimination steps of Gaussianelimination with partial pivoting,

.

Prove det (A) =

Proof:

[A] [A]-1 = [I]

If [A] is a nxn matrixand det (A) ¹ 0, what other statements are equivalent to it?

1. [A] is invertible.

2. [A]-1 exists.

3. [A] [X] = [C] has a unique solution.

4. [A] [X] = [0] solution is [X] =

5. [A] [A]-1 = [I] = [A]-1 [A].

Key Terms:

Naïve GaussElimination

Partial Pivoting

Determinant

 INTRODUCTION OF MATRIX ALGEBRA Topic Gaussian Elimination Summary Textbook notes on Gaussian Elimination Major All Majors of Engineering Authors Autar Kaw Last Revised December 26, 2014 Web Site http://numericalmethods.eng.usf.edu

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