Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![**Calculating the Adjugate, Determinant, and Inverse of a Matrix**
For the matrix \( A \) given by:
\[ A = \begin{bmatrix} 1 & 2 & -3 \\ 2 & 1 & -2 \\ 2 & -1 & 1 \end{bmatrix} \]
we need to perform the following calculations:
1. **Adjugate of \( A \) (adj\( A \))**
2. **Determinant of \( A \) (\( \det A \))**
3. **Inverse of \( A \) (\( A^{-1} \))**
**Steps to calculate each:**
1. **Determinant (\( \det A \)):**
\[
\det A = \begin{vmatrix} 1 & 2 & -3 \\ 2 & 1 & -2 \\ 2 & -1 & 1 \end{vmatrix}
\]
Expand the determinant using cofactor expansion along the first row:
\[
\det A = 1 \cdot \begin{vmatrix} 1 & -2 \\ -1 & 1 \end{vmatrix} - 2 \cdot \begin{vmatrix} 2 & -2 \\ 2 & 1 \end{vmatrix} + (-3) \cdot \begin{vmatrix} 2 & 1 \\ 2 & -1 \end{vmatrix}
\]
Calculating each 2x2 determinant:
\[
\begin{vmatrix} 1 & -2 \\ -1 & 1 \end{vmatrix} = (1)(1) - (-1)(-2) = 1 - 2 = -1
\]
\[
\begin{vmatrix} 2 & -2 \\ 2 & 1 \end{vmatrix} = (2)(1) - (2)(-2) = 2 - (-4) = 6
\]
\[
\begin{vmatrix} 2 & 1 \\ 2 & -1 \end{vmatrix} = (2)(-1) - (2)(1) = -2 - 2 = -4
\]
Substitute these back into the determinant formula:
\[
\det A = 1 \cdot (-1) - 2 \cdot](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1c145f92-040a-49d6-acdb-06a5c78bf220%2Fa1199216-576a-4fac-992f-d218bfc3760b%2F3rk5de7_processed.png&w=3840&q=75)
Transcribed Image Text:**Calculating the Adjugate, Determinant, and Inverse of a Matrix**
For the matrix \( A \) given by:
\[ A = \begin{bmatrix} 1 & 2 & -3 \\ 2 & 1 & -2 \\ 2 & -1 & 1 \end{bmatrix} \]
we need to perform the following calculations:
1. **Adjugate of \( A \) (adj\( A \))**
2. **Determinant of \( A \) (\( \det A \))**
3. **Inverse of \( A \) (\( A^{-1} \))**
**Steps to calculate each:**
1. **Determinant (\( \det A \)):**
\[
\det A = \begin{vmatrix} 1 & 2 & -3 \\ 2 & 1 & -2 \\ 2 & -1 & 1 \end{vmatrix}
\]
Expand the determinant using cofactor expansion along the first row:
\[
\det A = 1 \cdot \begin{vmatrix} 1 & -2 \\ -1 & 1 \end{vmatrix} - 2 \cdot \begin{vmatrix} 2 & -2 \\ 2 & 1 \end{vmatrix} + (-3) \cdot \begin{vmatrix} 2 & 1 \\ 2 & -1 \end{vmatrix}
\]
Calculating each 2x2 determinant:
\[
\begin{vmatrix} 1 & -2 \\ -1 & 1 \end{vmatrix} = (1)(1) - (-1)(-2) = 1 - 2 = -1
\]
\[
\begin{vmatrix} 2 & -2 \\ 2 & 1 \end{vmatrix} = (2)(1) - (2)(-2) = 2 - (-4) = 6
\]
\[
\begin{vmatrix} 2 & 1 \\ 2 & -1 \end{vmatrix} = (2)(-1) - (2)(1) = -2 - 2 = -4
\]
Substitute these back into the determinant formula:
\[
\det A = 1 \cdot (-1) - 2 \cdot
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