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.

Solution Steps

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.

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.

For an upper triangular matrix, the determinant is the product of its diagonal elements. Therefore, the determinant is calculated as follows: \[ \operatorname{det} = (-9) \times (-5) \times 9 \times 1 \times 7 \]

Calculate the product of the diagonal elements: \[ \operatorname{det} = (-9) \times (-5) \times 9 \times 1 \times 7 = 2835 \]

Final Answer