(a) The matrix A11 is 8 (b) The determinant of the matrix A₁1 is det A11 = (c) The matrix A12 is
(a) The matrix A11 is 8 (b) The determinant of the matrix A₁1 is det A11 = (c) The matrix A12 is
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![**Finding the Determinant of a Matrix Using Minors**
To find the determinant of the matrix \( A \):
\[
A = \begin{bmatrix} -6 & -3 & -9 \\ 6 & 8 & -1 \\ -7 & 7 & -1 \end{bmatrix}
\]
We will use the method of minors by expanding along the first row.
### Formula for Determinant Using Minors
The formula for \(\det A\) by expanding along the first row is:
\[
\det A = (-6 \times \det A_{11}) - (-3 \times \det A_{12}) + (-9 \times \det A_{13})
\]
where \(A_{11}\), \(A_{12}\), and \(A_{13}\) are appropriate submatrices of the matrix \(A\).
#### (a) The Matrix \(A_{11}\)
- **Diagram Area**: A placeholder for the 2x2 submatrix \( A_{11} \).
#### (b) The Determinant of the Matrix \(A_{11}\)
- **Formula Placeholder**: \(\det A_{11} =\) [Calculation to be done]
#### (c) The Matrix \(A_{12}\)
- **Diagram Area**: A placeholder for the 2x2 submatrix \( A_{12} \).
#### (d) The Determinant of the Matrix \(A_{12}\)
- **Formula Placeholder**: \(\det A_{12} =\) [Calculation to be done]
> By calculating each of these determinants and substituting back into the original formula, you can derive the overall determinant of matrix \(A\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F18395f57-75e3-4bdb-980a-ed8850989594%2Fcfaa71bc-09a7-49ee-af28-3752ee353504%2Fa4qnvr_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Finding the Determinant of a Matrix Using Minors**
To find the determinant of the matrix \( A \):
\[
A = \begin{bmatrix} -6 & -3 & -9 \\ 6 & 8 & -1 \\ -7 & 7 & -1 \end{bmatrix}
\]
We will use the method of minors by expanding along the first row.
### Formula for Determinant Using Minors
The formula for \(\det A\) by expanding along the first row is:
\[
\det A = (-6 \times \det A_{11}) - (-3 \times \det A_{12}) + (-9 \times \det A_{13})
\]
where \(A_{11}\), \(A_{12}\), and \(A_{13}\) are appropriate submatrices of the matrix \(A\).
#### (a) The Matrix \(A_{11}\)
- **Diagram Area**: A placeholder for the 2x2 submatrix \( A_{11} \).
#### (b) The Determinant of the Matrix \(A_{11}\)
- **Formula Placeholder**: \(\det A_{11} =\) [Calculation to be done]
#### (c) The Matrix \(A_{12}\)
- **Diagram Area**: A placeholder for the 2x2 submatrix \( A_{12} \).
#### (d) The Determinant of the Matrix \(A_{12}\)
- **Formula Placeholder**: \(\det A_{12} =\) [Calculation to be done]
> By calculating each of these determinants and substituting back into the original formula, you can derive the overall determinant of matrix \(A\).
![(c) The matrix \( A_{12} \) is
\[
\begin{bmatrix}
& \\
&
\end{bmatrix}
\]
(d) The determinant of the matrix \( A_{12} \) is
\[
\text{det. } A_{12} = \underline{\hspace{2cm}}
\]
(e) The matrix \( A_{13} \) is
\[
\begin{bmatrix}
& \\
&
\end{bmatrix}
\]
(f) The determinant of the matrix \( A_{13} \) is
\[
\text{det. } A_{13} = \underline{\hspace{2cm}}
\]
(g) The determinant of the matrix \( A \) is
\[
\text{det. } A = \underline{\hspace{2cm}}
\]
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Transcribed Image Text:(c) The matrix \( A_{12} \) is
\[
\begin{bmatrix}
& \\
&
\end{bmatrix}
\]
(d) The determinant of the matrix \( A_{12} \) is
\[
\text{det. } A_{12} = \underline{\hspace{2cm}}
\]
(e) The matrix \( A_{13} \) is
\[
\begin{bmatrix}
& \\
&
\end{bmatrix}
\]
(f) The determinant of the matrix \( A_{13} \) is
\[
\text{det. } A_{13} = \underline{\hspace{2cm}}
\]
(g) The determinant of the matrix \( A \) is
\[
\text{det. } A = \underline{\hspace{2cm}}
\]
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- Calculator
- Check Answer
Expert Solution
Step 1
given matrix
(a) find matrix
(b) determinant of matrix
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