For Problems 1-6, evaluate the determinant of the given matrix A by using (a) Definition 3.1.8 , (b) elementary row operations to reduce A to an upper triangular matrix, and (c) the Cofactor Expansion Theorem. A = [ 6 6 − 2 1 ] .

Question

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Answer 1

Textbook Question

Chapter 3.5, Problem 1AP

For Problems 1-6, evaluate the determinant of the given matrix A by using (a) Definition 3.1.8 , (b) elementary row operations to reduce A to an upper triangular matrix, and (c) the Cofactor Expansion Theorem.

A = [ 6 6 − 2 1 ] .

Expert Solution

To determine

a.

To explain:

The value of the determinant of matrix A=[66−21] using definition 3.1.8.

Answer to Problem 1AP

Solution:

The determinant of matrix A=[66−21] is 18.

Explanation of Solution

Given:

The matrix A=[66−21].

Approach:

By using the definition of determinants the value of general matrix of second order is a11a22−a12a21.

Calculation:

As, the value of determinant for general matrix of second order is a11a22−a12a21.

Then for matrix A the value of det(A)=6×1−(6×(−2)).

So, det(A)=18.

Expert Solution

To determine

b.

The value of the determinant of matrix A=[66−21] using elementary row operations to reduce A to an upper triangular matrix.

Answer to Problem 1AP

Solution:

The determinant of matrix A=[66−21] is 18 using elementary row operations to reduce A to an upper triangular matrix.

Explanation of Solution

Given:

The matrix A=[66−21].

Approach:

By converting the given matrix to upper triangular matrix in which the entries below diagonal elements are zero and then taking the product of diagonal elements of the equivalent matrix gives the determinant of the matrix.

Calculation:

Firstly apply the row operation as adding one third of the first row to the second row it gives A∼[6603].

Thus, det(A)=6×3.

That is, det(A)=18.

Expert Solution

To determine

c.

The value of the determinant of matrix A=[66−21] using cofactor expansion theorem.

Answer to Problem 1AP

Solution:

The determinant of matrix A=[66−21] is 18 using cofactor expansion theorem.

Explanation of Solution

Given:

The matrix A=[66−21].

Approach:

According to cofactor expansion theorem the determinant of general second order matrix is given by a11C11+a12C12 where C11=|a22|,C12=−|a21|

Calculation:

So, by the expansion theorem the det(A)=6×|1|+6×(−|−2|).

Thus, det(A)=18.

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Answer 2

Textbook Question

Chapter 3.5, Problem 1AP

For Problems 1-6, evaluate the determinant of the given matrix A by using (a) Definition 3.1.8 , (b) elementary row operations to reduce A to an upper triangular matrix, and (c) the Cofactor Expansion Theorem.

A = [ 6 6 − 2 1 ] .

Expert Solution

To determine

a.

To explain:

The value of the determinant of matrix A=[66−21] using definition 3.1.8.

Answer to Problem 1AP

Solution:

The determinant of matrix A=[66−21] is 18.

Explanation of Solution

Given:

The matrix A=[66−21].

Approach:

By using the definition of determinants the value of general matrix of second order is a11a22−a12a21.

Calculation:

As, the value of determinant for general matrix of second order is a11a22−a12a21.

Then for matrix A the value of det(A)=6×1−(6×(−2)).

So, det(A)=18.

Expert Solution

To determine

b.

The value of the determinant of matrix A=[66−21] using elementary row operations to reduce A to an upper triangular matrix.

Answer to Problem 1AP

Solution:

The determinant of matrix A=[66−21] is 18 using elementary row operations to reduce A to an upper triangular matrix.

Explanation of Solution

Given:

The matrix A=[66−21].

Approach:

By converting the given matrix to upper triangular matrix in which the entries below diagonal elements are zero and then taking the product of diagonal elements of the equivalent matrix gives the determinant of the matrix.

Calculation:

Firstly apply the row operation as adding one third of the first row to the second row it gives A∼[6603].

Thus, det(A)=6×3.

That is, det(A)=18.

Expert Solution

To determine

c.

The value of the determinant of matrix A=[66−21] using cofactor expansion theorem.

Answer to Problem 1AP

Solution:

The determinant of matrix A=[66−21] is 18 using cofactor expansion theorem.

Explanation of Solution

Given:

The matrix A=[66−21].

Approach:

According to cofactor expansion theorem the determinant of general second order matrix is given by a11C11+a12C12 where C11=|a22|,C12=−|a21|

Calculation:

So, by the expansion theorem the det(A)=6×|1|+6×(−|−2|).

Thus, det(A)=18.

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Answer 3

Textbook Question

Chapter 3.5, Problem 1AP

For Problems 1-6, evaluate the determinant of the given matrix A by using (a) Definition 3.1.8 , (b) elementary row operations to reduce A to an upper triangular matrix, and (c) the Cofactor Expansion Theorem.

A = [ 6 6 − 2 1 ] .

Expert Solution

To determine

a.

To explain:

The value of the determinant of matrix A=[66−21] using definition 3.1.8.

Answer to Problem 1AP

Solution:

The determinant of matrix A=[66−21] is 18.

Explanation of Solution

Given:

The matrix A=[66−21].

Approach:

By using the definition of determinants the value of general matrix of second order is a11a22−a12a21.

Calculation:

As, the value of determinant for general matrix of second order is a11a22−a12a21.

Then for matrix A the value of det(A)=6×1−(6×(−2)).

So, det(A)=18.

Expert Solution

To determine

b.

The value of the determinant of matrix A=[66−21] using elementary row operations to reduce A to an upper triangular matrix.

Answer to Problem 1AP

Solution:

The determinant of matrix A=[66−21] is 18 using elementary row operations to reduce A to an upper triangular matrix.

Explanation of Solution

Given:

The matrix A=[66−21].

Approach:

By converting the given matrix to upper triangular matrix in which the entries below diagonal elements are zero and then taking the product of diagonal elements of the equivalent matrix gives the determinant of the matrix.

Calculation:

Firstly apply the row operation as adding one third of the first row to the second row it gives A∼[6603].

Thus, det(A)=6×3.

That is, det(A)=18.

Expert Solution

To determine

c.

The value of the determinant of matrix A=[66−21] using cofactor expansion theorem.

Answer to Problem 1AP

Solution:

The determinant of matrix A=[66−21] is 18 using cofactor expansion theorem.

Explanation of Solution

Given:

The matrix A=[66−21].

Approach:

According to cofactor expansion theorem the determinant of general second order matrix is given by a11C11+a12C12 where C11=|a22|,C12=−|a21|

Calculation:

So, by the expansion theorem the det(A)=6×|1|+6×(−|−2|).

Thus, det(A)=18.

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Answer 4

Textbook Question

Chapter 3.5, Problem 1AP

For Problems 1-6, evaluate the determinant of the given matrix A by using (a) Definition 3.1.8 , (b) elementary row operations to reduce A to an upper triangular matrix, and (c) the Cofactor Expansion Theorem.

A = [ 6 6 − 2 1 ] .

Expert Solution

To determine

a.

To explain:

The value of the determinant of matrix A=[66−21] using definition 3.1.8.

Answer to Problem 1AP

Solution:

The determinant of matrix A=[66−21] is 18.

Explanation of Solution

Given:

The matrix A=[66−21].

Approach:

By using the definition of determinants the value of general matrix of second order is a11a22−a12a21.

Calculation:

As, the value of determinant for general matrix of second order is a11a22−a12a21.

Then for matrix A the value of det(A)=6×1−(6×(−2)).

So, det(A)=18.

Expert Solution

To determine

b.

The value of the determinant of matrix A=[66−21] using elementary row operations to reduce A to an upper triangular matrix.

Answer to Problem 1AP

Solution:

The determinant of matrix A=[66−21] is 18 using elementary row operations to reduce A to an upper triangular matrix.

Explanation of Solution

Given:

The matrix A=[66−21].

Approach:

By converting the given matrix to upper triangular matrix in which the entries below diagonal elements are zero and then taking the product of diagonal elements of the equivalent matrix gives the determinant of the matrix.

Calculation:

Firstly apply the row operation as adding one third of the first row to the second row it gives A∼[6603].

Thus, det(A)=6×3.

That is, det(A)=18.

Expert Solution

To determine

c.

The value of the determinant of matrix A=[66−21] using cofactor expansion theorem.

Answer to Problem 1AP

Solution:

The determinant of matrix A=[66−21] is 18 using cofactor expansion theorem.

Explanation of Solution

Given:

The matrix A=[66−21].

Approach:

According to cofactor expansion theorem the determinant of general second order matrix is given by a11C11+a12C12 where C11=|a22|,C12=−|a21|

Calculation:

So, by the expansion theorem the det(A)=6×|1|+6×(−|−2|).

Thus, det(A)=18.

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Answer 5

Textbook Question

Answer 6

Textbook Question

Answer 7

Expert Solution

To determine

a.

To explain:

The value of the determinant of matrix A=[66−21] using definition 3.1.8.

Answer to Problem 1AP

Solution:

The determinant of matrix A=[66−21] is 18.

Explanation of Solution

Given:

The matrix A=[66−21].

Approach:

By using the definition of determinants the value of general matrix of second order is a11a22−a12a21.

Calculation:

As, the value of determinant for general matrix of second order is a11a22−a12a21.

Then for matrix A the value of det(A)=6×1−(6×(−2)).

So, det(A)=18.

Expert Solution

To determine

b.

The value of the determinant of matrix A=[66−21] using elementary row operations to reduce A to an upper triangular matrix.

Answer to Problem 1AP

Solution:

The determinant of matrix A=[66−21] is 18 using elementary row operations to reduce A to an upper triangular matrix.

Explanation of Solution

Given:

The matrix A=[66−21].

Approach:

By converting the given matrix to upper triangular matrix in which the entries below diagonal elements are zero and then taking the product of diagonal elements of the equivalent matrix gives the determinant of the matrix.

Calculation:

Firstly apply the row operation as adding one third of the first row to the second row it gives A∼[6603].

Thus, det(A)=6×3.

That is, det(A)=18.

Expert Solution

To determine

c.

The value of the determinant of matrix A=[66−21] using cofactor expansion theorem.

Answer to Problem 1AP

Solution:

The determinant of matrix A=[66−21] is 18 using cofactor expansion theorem.

Explanation of Solution

Given:

The matrix A=[66−21].

Approach:

According to cofactor expansion theorem the determinant of general second order matrix is given by a11C11+a12C12 where C11=|a22|,C12=−|a21|

Calculation:

So, by the expansion theorem the det(A)=6×|1|+6×(−|−2|).

Thus, det(A)=18.

Answer 8

Expert Solution

To determine

a.

To explain:

The value of the determinant of matrix A=[66−21] using definition 3.1.8.

Answer to Problem 1AP

Solution:

The determinant of matrix A=[66−21] is 18.

Explanation of Solution

Given:

The matrix A=[66−21].

Approach:

By using the definition of determinants the value of general matrix of second order is a11a22−a12a21.

Calculation:

As, the value of determinant for general matrix of second order is a11a22−a12a21.

Then for matrix A the value of det(A)=6×1−(6×(−2)).

So, det(A)=18.

Answer 9

Expert Solution

Answer 10

Expert Solution

Answer 11

Expert Solution

Answer 12

To determine

a.

To explain:

The value of the determinant of matrix A=[66−21] using definition 3.1.8.

Answer 13

Answer to Problem 1AP

Solution:

The determinant of matrix A=[66−21] is 18.

Answer 14

Explanation of Solution

Given:

The matrix A=[66−21].

Approach:

By using the definition of determinants the value of general matrix of second order is a11a22−a12a21.

Calculation:

As, the value of determinant for general matrix of second order is a11a22−a12a21.

Then for matrix A the value of det(A)=6×1−(6×(−2)).

So, det(A)=18.

Answer 15

Expert Solution

To determine

b.

The value of the determinant of matrix A=[66−21] using elementary row operations to reduce A to an upper triangular matrix.

Answer to Problem 1AP

Solution:

The determinant of matrix A=[66−21] is 18 using elementary row operations to reduce A to an upper triangular matrix.

Explanation of Solution

Given:

The matrix A=[66−21].

Approach:

By converting the given matrix to upper triangular matrix in which the entries below diagonal elements are zero and then taking the product of diagonal elements of the equivalent matrix gives the determinant of the matrix.

Calculation:

Firstly apply the row operation as adding one third of the first row to the second row it gives A∼[6603].

Thus, det(A)=6×3.

That is, det(A)=18.

Answer 16

Expert Solution

Answer 17

Expert Solution

Answer 18

Expert Solution

Answer 19

To determine

b.

The value of the determinant of matrix A=[66−21] using elementary row operations to reduce A to an upper triangular matrix.

Answer 20

Answer to Problem 1AP

Solution:

The determinant of matrix A=[66−21] is 18 using elementary row operations to reduce A to an upper triangular matrix.

Answer 21

Explanation of Solution

Given:

The matrix A=[66−21].

Approach:

By converting the given matrix to upper triangular matrix in which the entries below diagonal elements are zero and then taking the product of diagonal elements of the equivalent matrix gives the determinant of the matrix.

Calculation:

Firstly apply the row operation as adding one third of the first row to the second row it gives A∼[6603].

Thus, det(A)=6×3.

That is, det(A)=18.

Answer 22

Expert Solution

To determine

c.

The value of the determinant of matrix A=[66−21] using cofactor expansion theorem.

Answer to Problem 1AP

Solution:

The determinant of matrix A=[66−21] is 18 using cofactor expansion theorem.

Explanation of Solution

Given:

The matrix A=[66−21].

Approach:

According to cofactor expansion theorem the determinant of general second order matrix is given by a11C11+a12C12 where C11=|a22|,C12=−|a21|

Calculation:

So, by the expansion theorem the det(A)=6×|1|+6×(−|−2|).

Thus, det(A)=18.

Answer 23

Expert Solution

Answer 24

Expert Solution

Answer 25

Expert Solution

Answer 26

To determine

c.

The value of the determinant of matrix A=[66−21] using cofactor expansion theorem.

Answer 27

Answer to Problem 1AP

Solution:

The determinant of matrix A=[66−21] is 18 using cofactor expansion theorem.

Answer 28

Explanation of Solution

Given:

The matrix A=[66−21].

Approach:

According to cofactor expansion theorem the determinant of general second order matrix is given by a11C11+a12C12 where C11=|a22|,C12=−|a21|

Calculation:

So, by the expansion theorem the det(A)=6×|1|+6×(−|−2|).

Thus, det(A)=18.

Answer 29

Ch. 3.1 - True-False Review For items (a)(j), decide if the...Ch. 3.1 - True-False Review For items (a)(j), decide if the...Ch. 3.1 - True-False Review For items a-j, decide if the...Ch. 3.1 - For Problems 1-6, determine the number of...Ch. 3.1 - For Problems 1-6, determine the number of...Ch. 3.1 - For Problems 1-6, determine the number of...Ch. 3.1 - For Problems 1-6, determine the number of...Ch. 3.1 - Prob. 5P Ch. 3.1 - Prob. 6P Ch. 3.1 - Use Definition 3.1.8 to derive the general...

For Problems 1-6, evaluate the determinant of the given matrix A by using (a) Definition 3.1.8 , (b) elementary row operations to reduce A to an upper triangular matrix, and (c) the Cofactor Expansion Theorem. A = [ 6 6 − 2 1 ] .

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