Chapter 5: Chapter 5

Problem 139

Evaluate det $\mathrm{A}$ where: $A=\left|\begin{array}{rrr}0 & 1 & 5 \\ 3 & -6 & 9 \\ 2 & 6 & 1\end{array}\right|$

Short Answer

Expert verified

The determinant of matrix $A = \begin{pmatrix} 0 & 1 & 5 \\ 3 & -6 & 9 \\ 2 & 6 & 1 \end{pmatrix}$ can be found using the expansion by minors. First, expand along the first row: $\det(A) = 0 \cdot \det(M_{11}) - 1 \cdot \det(M_{12}) + 5 \cdot \det(M_{13})$, where the minor matrices are $M_{11} = \begin{pmatrix}-6 & 9 \\ 6 & 1 \end{pmatrix}$, $M_{12} = \begin{pmatrix} 3 & 9 \\ 2 & 1 \end{pmatrix}$, and $M_{13} = \begin{pmatrix} 3 & -6 \\ 2 & 6 \end{pmatrix}$. Then, find the determinant of each minor matrix: $\det(M_{11}) = -60$, $\det(M_{12}) = -15$, and $\det(M_{13}) = 24$. Substituting these values back into the determinant formula, we get $\det(A) = 0 + 15 + 120 = 135$.

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Step by step solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Write down the given matrix

First, let's write down the given matrix A: \[ A=\begin{pmatrix} 0 & 1 & 5 \\ 3 & -6 & 9 \\ 2 & 6 & 1 \end{pmatrix} \]

Step 2: Choose a row or column to expand along

Next, choose a row or column to expand the determinant along. In this case, we'll choose the first row since it contains a 0, so it will simplify our calculations. \[ \det(A) = 0 \cdot \det(M_{11}) - 1 \cdot \det(M_{12}) + 5 \cdot \det(M_{13}) \]

Step 3: Calculate Minor matrices

Now, we have to find the 2x2 minor matrices (Mij) where i=1 and j= 1,2,3. \[ M_{11} = \begin{pmatrix} -6 & 9 \\ 6 & 1 \end{pmatrix}, \ M_{12} = \begin{pmatrix} 3 & 9 \\ 2 & 1 \end{pmatrix}, \ M_{13} = \begin{pmatrix} 3 & -6 \\ 2 & 6 \end{pmatrix} \]

Step 4: Find the determinant of each minor

Next, we'll find the determinant of each minor matrix: \[ \det(M_{11}) = (-6)(1) - (9)(6) = -60 \\ \det(M_{12}) = (3)(1) - (9)(2) = -15 \\ \det(M_{13}) = (3)(6) - (-6)(2) = 24 \]

Step 5: Substitute the values into the determinant formula

Substitute the calculated determinant values of minor matrices back into the previous formula: \[ \det(A) = 0 \cdot (-60) - 1 \cdot (-15) + 5 \cdot (24) \\ \]

Step 6: Calculate the determinant of matrix A

Finally, we'll calculate the determinant of matrix A: \[ \det(A) = 0 + 15 + 120 = 135 \\ \] The determinant of matrix A is 135.

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Chapter 5: Chapter 5

Evaluate det \(\mathrm{A}\) where: $A=\left|\begin{array}{rrr}0 & 1 & 5 \\ 3 & -6 & 9 \\ 2 & 6 & 1\end{array}\right|$

Short Answer

Step by step solution

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Step 1: Write down the given matrix

Step 2: Choose a row or column to expand along

Step 3: Calculate Minor matrices

Step 4: Find the determinant of each minor

Step 5: Substitute the values into the determinant formula

Step 6: Calculate the determinant of matrix A

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Chapter 1

Chapter 2

Chapter 3

Chapter 4

Chapter 6

Chapter 7

Chapter 8

Chapter 9

Chapter 10

Chapter 11

Chapter 12

Chapter 13

Chapter 14

Chapter 15

Chapter 16

Chapter 17

Chapter 18

Chapter 19

Chapter 20

Chapter 21

Chapter 22

Chapter 23

Chapter 24

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