Algebra: A Combined Approach (4th Edition)

Published by Pearson
ISBN 10: 0321726391
ISBN 13: 978-0-32172-639-1

Appendix E - Exercise Set - Page 1005: 38

Answer

$0$

Work Step by Step

Let us consider a $ 3\times 3$ determinant with all the elements in the first column be $0$. The first column of the array of signs is $ + , - , +$ . $$\begin{vmatrix} 0 & a & d\\ 0 & b & c\\ 0 & e&f \end{vmatrix}=0\begin{vmatrix} b & c \\ e & f \end{vmatrix}-0\begin{vmatrix} a& d \\ e& f \end{vmatrix}+0\begin{vmatrix} a & d \\ b& c \end{vmatrix}$$ $$=0(bf-ec)-0(af-de)+0(ac-db)=0-0+0=0$$ Note that the determinant of the matrix will be the same regardless of the column you select to expand by. The result is the same if the determinant has $n$ rows and $n$ columns.
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