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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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