Questions: Question 7 1 pt Find the determinant of the matrix. det[[-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]] = Question 8 1 pt Find 3 × 3 lower triangular matrix with determinant -7.

Question 7
1 pt

Find the determinant of the matrix.

det[[-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]] =

Question 8
1 pt

Find 3 × 3 lower triangular matrix with determinant -7.
Transcript text: Question 7 1 pt Find the determinant of the matrix. \[ \operatorname{det}\left[\begin{array}{ccccc} -9 & 0 & 0 & 0 & 0 \\ 7 & -5 & 0 & 0 & 0 \\ -3 & 1 & 9 & 0 & 0 \\ 1 & 8 & -1 & 1 & 0 \\ 4 & -5 & -4 & 5 & 7 \end{array}\right]= \] $\square$ Question Help: Video Written Example Submit Question Question 8 1 pt Find $3 \times 3$ lower triangular matrix with determinant -7 . $\square$ $\square$ $\square$ $\square$ $\square$ $\square$ $\square$ $\square$ $\square$ \[ Y \] Question Help: Video Written Example Submit All Parts
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Solution

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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 \(\boxed{2835}\).

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