Stitz-Zeager_College_Algebra_e-book

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Unformatted text preview: t be 1. That is, A’s reduced row echelon form is In , and A is invertible. Conversely, if A is invertible, then A can be transformed into In using row operations. Since det (In ) = 1 = 0, our same logic implies det(A) = 0. Basically, we have established that the determinant determines whether or not the matrix A is invertible.4 It is worth noting that when we first introduced the notion of a matrix inverse, it was in the context of solving a linear matrix equation. In effect, we were trying to ‘divide’ both sides of the matrix equation AX = B by the matrix A. Just like we cannot divide a real number by 0, Theorem 8.7 tells us we cannot ‘divide’ by a matrix whose determinant is 0. We also know that if the coefficient matrix of a system of linear equations is invertible, then system is consistent and independent. It follows, then, that if the determinant of said coefficient is not zero, the system is consistent and independent. 8.5.2 Cramer’s Rule and Matrix Adjoints In this section, we introduce a theorem which enables us to solve a system of linear equations by means of determinants only. As usual, the theorem is stated in full generality, using numbered unknowns x1 , x2 , etc., instead of the more familiar letter...
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This note was uploaded on 05/03/2013 for the course MATH Algebra taught by Professor Wong during the Fall '13 term at Chicago Academy High School.

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