Transcript text: Question 7
1 pt
Find the determinant of the matrix.
\[
\operatorname{det}\left[\begin{array}{ccccc}
-9 & 0 & 0 & 0 & 0 \\
7 & -5 & 0 & 0 & 0 \\
-3 & 1 & 9 & 0 & 0 \\
1 & 8 & -1 & 1 & 0 \\
4 & -5 & -4 & 5 & 7
\end{array}\right]=
\]
$\square$
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Question 8
1 pt
Find $3 \times 3$ lower triangular matrix with determinant -7 .
$\square$
$\square$
$\square$
$\square$
$\square$
$\square$
$\square$
$\square$
$\square$
\[
Y
\]
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Solution
Solution Steps
Solution Approach
To find the determinant of a 5x5 matrix, we can use a property of triangular matrices. If a matrix is upper or lower triangular, its determinant is the product of its diagonal elements. In this case, the matrix is upper triangular because all elements below the main diagonal are zero. Therefore, the determinant is the product of the diagonal elements.
Step 1: Identify the Matrix Type
The given matrix is:
\[
\begin{bmatrix}
-9 & 0 & 0 & 0 & 0 \\
7 & -5 & 0 & 0 & 0 \\
-3 & 1 & 9 & 0 & 0 \\
1 & 8 & -1 & 1 & 0 \\
4 & -5 & -4 & 5 & 7
\end{bmatrix}
\]
This matrix is an upper triangular matrix because all elements below the main diagonal are zero.
Step 2: Calculate the Determinant
For an upper triangular matrix, the determinant is the product of its diagonal elements. Therefore, the determinant is calculated as follows:
\[
\operatorname{det} = (-9) \times (-5) \times 9 \times 1 \times 7
\]
Step 3: Compute the Product
Calculate the product of the diagonal elements:
\[
\operatorname{det} = (-9) \times (-5) \times 9 \times 1 \times 7 = 2835
\]
Final Answer
The determinant of the matrix is \(\boxed{2835}\).