Chapter Nine — Matrix Operations and Calculations
The Humongous Book of Algebra Problems
192
9.25 Calculate the matrix product below.
The product of two 3 × 3 matrices is a 3 × 3 matrix.
Calculating Determinants
Values dened for square matrices only
9.26 Given matrix A defined below, calculate .
The determinant of matrix A, written “ ” or “det (A),” is a real number value
defined when A is a square matrix. The determinant of a 2 × 2 matrix can be
calculated using the below shortcut formula.
Set a = 3, b = –2, c = 1, and d = 9.
A square matrix
has the same number of
rows and columns.
Start at
the upper-left
corner and multiply
diagonally down.
Then, subtract what
you get when you
start at the lower-
left corner and
multiply diagonally
up.
Chapter Nine — Matrix Operations and Calculations
The Humongous Book of Algebra Problems
193
9.27 Given matrix B defined below, calculate .
Apply the shortcut formula from Problem 9.26 to calculate the determinant of
the 2 × 2 matrix.
9.28 Given matrix C defined below, calculate .
Apply the shortcut formula from Problem 9.26 to calculate the determinant of
the 2 × 2 matrix.
9.29 Given matrix D defined below, calculate .
Problem 9.26 describes a shortcut used to calculate the determinant of a
2 × 2 matrix. It requires you to subtract a product of elements along an upward
diagonal from a product of elements along a downward diagonal.
Chapter Nine — Matrix Operations and Calculations
The Humongous Book of Algebra Problems
194
The shortcut technique for calculating the matrix of a 3 × 3 determinant also
involves the difference of products along diagonals, but it requires you to
construct a 3 × 5 matrix. The first three columns consist of the original matrix,
column four is a copy of column one, and column five is a copy of column two.
Beginning with element d
11
= 2, calculate the products along the diagonal and
sum the results. Do the same with elements d
12
= –5 and d
13
= 1.
Now multiply along the upward diagonals that start at d
31
= 0, d
32
= –1, and d
33
= 8
and add the products.
The determinant of the 3 × 3 matrix is the difference of the downward and
upward diagonal sums.
9.30 Given matrix E defined here, calculate det(E).
Construct a 3 × 5 matrix by duplicating the first two columns of the matrix, as
directed by Problem 9.29.
Stick copies
of the rst two
columns on the right
side of the 3 × 3
matrix to create a
3 × 5 matrix you’ll use
to calculate the
determinant.
Just like a 2 × 2
determinant
is equal to the
difference of a
downward and an
upward diagonal.
Det(E) and
both mean the same
thing: the determinant
of matrix E.
Chapter Nine — Matrix Operations and Calculations
The Humongous Book of Algebra Problems
195
Calculate the determinant by adding the products of the elements in the
downward diagonals beginning at e
11
= 1, e
12
= 3, and e
13
= –6 and then subtracting
the products of the elements in the upward diagonals beginning at e
31
= 4,
, and e
33
= 5.
Note: Problems 9.31–9.35 refer to matrix A defined below.
9.31 Calculate the minor M
11
of A.
The minor M
ij
of a matrix is the determinant of the matrix created by removing
the ith row and the jth column. This problem directs you to calculate M
11
, so
eliminate the first row and the first column from the matrix and calculate the
determinant of the resulting 2 × 2 matrix.
Apply the shortcut formula for calculating 2 × 2 matrices presented in Problem
9.26.
Note: Problems 9.31–9.35 refer to matrix A defined below.
9.32 Calculate the cofactor C
11
of A.
The cofactor C
ij
of a matrix is computed according to the following formula:
C
ij
= (–1)
i + j
⋅ M
ij
. Therefore, the cofactor is equal to the corresponding minor
multiplied either by +1 or –1, depending upon the value of the expression i + j.
According to Problem 9.31, M
11
= 84. Substitute M
11
, i = 1, and j = 1 into the
cofactor formula.
Drop the
rst row (which
contains 3 4 –1)
and the rst column
(which contains 3 0
1) and you’re left
with this.
You’re calcu-
lating C
11
. The rst
little number next to
C is i and the second
little number is j.
They’re both equal
to 1 here, so
i = 1 and j = 1.
Chapter Nine — Matrix Operations and Calculations
The Humongous Book of Algebra Problems
196
Note: Problems 9.31–9.35 refer to matrix A defined below.
9.33 Calculate the cofactor C
23
of A.
Begin by calculating M
23
, the determinant of the matrix created by removing
the second row and the third column of A.
Substitute M
23
= –22, i = 2, and j = 3 into the cofactor formula.
Note: Problems 9.31–9.35 refer to matrix A defined below.
9.34 Calculate by performing a cofactor expansion along the third row of A.
Though you can calculate the determinant of this 3 × 3 matrix using the
shortcut described in Problem 9.29, the cofactor expansion technique can be
applied to any square matrix. Multiply each of the elements in a single row or
column of the matrix by the corresponding cofactor and add the results.
This problem directs you to expand along the third row, but the results would
be the same if you chose to expand along either of the other rows or any of the
columns.
It might
help to temp-
orarily cross row
2 and column 3
out of the matrix to
visualize the 2 × 2
matrix you’ll end up
with:
As you’ll see in the
next problem, when you
expand along a column
you will get the same
answer.
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