Chapter Nine — Matrix Operations and Calculations
The Humongous Book of Algebra Problems
197
Calculate the minors M
31
, M
32
, and M
33
using the technique described in
Problem 9.31.
Next, calculate the corresponding cofactors: C
31
, C
32
, and C
33
.
Multiply each element of the third row by the corresponding cofactor. The sum
of those products is the determinant of matrix A.
Note: Problems 9.31–9.35 refer to matrix A defined below.
9.35 Verify the solution to Problem 9.34 by performing a cofactor expansion along
the first column of A.
Note that a
21
= 0, so there is no need to calculate C
21
for the cofactor expansion.
According to Problem 9.32, C
11
= 84, and according to Problem 9.34, C
31
= 38.
No matter
what C
21
is, during
the expansion it gets
multiplied by a
21
= 0,
and the result will
be 0.
Chapter Nine — Matrix Operations and Calculations
The Humongous Book of Algebra Problems
198
9.36 According to Problem 9.30, the determinant of matrix E (defined below) is
412. Perform a cofactor expansion to verify the determinant.
To perform a cofactor expansion along the first row, calculate M
11
, M
12
, and M
13
.
Calculate the corresponding cofactors.
Multiply the elements of the first row by the corresponding cofactors and add
the results.
9.37 Given matrix B defined below, calculate .
Because b
13
= 0, it is most efficient to expand along either the first row or the
third column. To expand along the third column of B, calculate M
23
, M
33
, and
M
43
. Apply the shortcut technique described in Problem 9.29 to calculate the
3 × 3 determinants.
The book
shows you
how to expand
along the rst
row, but you’ll get
the same answer
with either of the
other two rows or if
you expanded one
of the columns
instead.
Problem
9.35 explains
that a 0 in the
row or column you’re
expanding means
that you don’t
have to calculate
the corresponding
cofactor, and that
means a little
less work.
Chapter Nine — Matrix Operations and Calculations
The Humongous Book of Algebra Problems
199
Calculate the corresponding cofactors.
Multiply the elements of the third column by the corresponding cofactors and
sum the results.
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