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The Humongous Book of Algebra Problems by W. Michael Kelley

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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 dened 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 youre left
with this.
Youre 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.

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