**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

To find the adjoint determinant and inverse of a matrix

Answered: Calculate adj A, det A, and hence A¯',…

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Calculate adj A, det A, and hence A¯', for the matrix A=| 2 1 -2 -1 1

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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Question

**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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