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.

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