Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN: 9780134463216
Author: Robert F. Blitzer
Publisher: PEARSON
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### Finding Values of \( a \) for Determinant of Matrix \( A \) In this exercise, we are given a matrix \(A\) and tasked with finding the values of the variable \( a \) that make the determinant of \( A \) equal to zero. The matrix \( A \) is defined as: \[ A = \begin{bmatrix} a & 5 & 7 \\ a & -2 & 4 \\ 4 & 0 & a \\ \end{bmatrix} \] #### Problem Statement Find all values of \( a \) that make \(|A| = 0\). Give your answer as a comma-separated list. #### Input Box: Values of \( a \): [ __________ ] ### Instructions To solve this problem, you need to calculate the determinant of the given 3x3 matrix \( A \). The determinant of a 3x3 matrix: \[ \begin{vmatrix} a & 5 & 7 \\ a & -2 & 4 \\ 4 & 0 & a \\ \end{vmatrix} \] is found using the formula: \[ \text{det}(A) = a \cdot \text{det} \begin{bmatrix} -2 & 4 \\ 0 & a \end{bmatrix} - 5 \cdot \text{det} \begin{bmatrix} a & 4 \\ 4 & a \end{bmatrix} + 7 \cdot \text{det} \begin{bmatrix} a & -2 \\ 4 & 0 \end{bmatrix} \] By evaluating the above, we can find the values of \( a \) that make the determinant zero. Enter the resulting values of \( a \) in the input box provided as a comma-separated list.
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Transcribed Image Text:### Finding Values of \( a \) for Determinant of Matrix \( A \) In this exercise, we are given a matrix \(A\) and tasked with finding the values of the variable \( a \) that make the determinant of \( A \) equal to zero. The matrix \( A \) is defined as: \[ A = \begin{bmatrix} a & 5 & 7 \\ a & -2 & 4 \\ 4 & 0 & a \\ \end{bmatrix} \] #### Problem Statement Find all values of \( a \) that make \(|A| = 0\). Give your answer as a comma-separated list. #### Input Box: Values of \( a \): [ __________ ] ### Instructions To solve this problem, you need to calculate the determinant of the given 3x3 matrix \( A \). The determinant of a 3x3 matrix: \[ \begin{vmatrix} a & 5 & 7 \\ a & -2 & 4 \\ 4 & 0 & a \\ \end{vmatrix} \] is found using the formula: \[ \text{det}(A) = a \cdot \text{det} \begin{bmatrix} -2 & 4 \\ 0 & a \end{bmatrix} - 5 \cdot \text{det} \begin{bmatrix} a & 4 \\ 4 & a \end{bmatrix} + 7 \cdot \text{det} \begin{bmatrix} a & -2 \\ 4 & 0 \end{bmatrix} \] By evaluating the above, we can find the values of \( a \) that make the determinant zero. Enter the resulting values of \( a \) in the input box provided as a comma-separated list.
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