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8_ I Matrices and Determinants
When expanding by cofactors, you do not need to find cofactors of zero entries,
because zero times its cofactor is zero.
afa = (0)C,j = 0
Thus, the row (or column) containing the most zeros is usually the best choice
for expansion by cofactors. This is demonstrated in the next example.
EXAMPLE 4
^
Th e Determinan t o f a Matri x o f Orde r 4 x 4
Find the determinant of
A =
1
-1
0
3
—z
3
0
0
0
1
2
4
0
2
3
2
Solution
After inspecting this matrix, you can see that three of the entries in the third
column are zeros. Thus, you can eliminate some of the work in the expansion
by using the third column.
\A\ = 3(C13) + 0(C23) + 0(C33) + 0(Q 3 )
Because C23, C33, and C43 have zero coefficients, you need only find the
cofactor C ]3 . To do this, delete the first row and third column of A and evaluate the determinant of the resulting matrix.
Ct3 = ("I)'
-1
= 0
3
Study
Although most graphing utilities
can calculate the determinant of a
square matrix, it is also important
to know how to calculate them by
hand.
1
0
3
+3
1
2
4
1
2
4
2
3
2
2
3
2
Delete 1s t ro w an d 3r d column .
Simplify.
Expanding by cofactors in the second row yields the following.
+ 2( -
-1
3
-1
3
= 0
= 5
Thus, you obtain
\A\ = 3C, 3 = 3(5) = 15.
Note Try using a graphing utility to confirm the result of Example 4.
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