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Let A be a 3-by-3 matrix whose determinant is -5. Use properties of determinants to evaluate the following: det(2A) = det (A²) = det (A-1) =

Let A be a 3-by-3 matrix whose determinant is -5. Use properties of determinants to evaluate the following: det(2A) = det (A²) = det (A-1) =

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Matrix Determinants Evaluation** Let \( A \) be a 3-by-3 matrix whose determinant is \(-5\). Use properties of determinants to evaluate the following: 1. \( \text{det}(2A) = \) 2. \( \text{det}(A^2) = \) 3. \( \text{det}(A^{-1}) = \) **Properties of Determinants:** 1. **Scalar Multiplication:** For a scalar \( c \) and an \( n \times n \) matrix \( A \), the determinant of \( cA \) is \( c^n \times \text{det}(A) \). 2. **Powers of a Matrix:** The determinant of \( A^k \) is \( (\text{det}(A))^k \). 3. **Inverse of a Matrix:** The determinant of the inverse of \( A \), \( A^{-1} \), is \( 1/\text{det}(A) \). Using these properties, you can fill in the blanks for the determinant evaluations: - \( \text{det}(2A) = 2^3 \times (-5) \) - \( \text{det}(A^2) = (-5)^2 \) - \( \text{det}(A^{-1}) = \frac{1}{-5} \) Completing these calculations will provide the evaluated determinants for each expression.
Transcribed Image Text:**Matrix Determinants Evaluation** Let \( A \) be a 3-by-3 matrix whose determinant is \(-5\). Use properties of determinants to evaluate the following: 1. \( \text{det}(2A) = \) 2. \( \text{det}(A^2) = \) 3. \( \text{det}(A^{-1}) = \) **Properties of Determinants:** 1. **Scalar Multiplication:** For a scalar \( c \) and an \( n \times n \) matrix \( A \), the determinant of \( cA \) is \( c^n \times \text{det}(A) \). 2. **Powers of a Matrix:** The determinant of \( A^k \) is \( (\text{det}(A))^k \). 3. **Inverse of a Matrix:** The determinant of the inverse of \( A \), \( A^{-1} \), is \( 1/\text{det}(A) \). Using these properties, you can fill in the blanks for the determinant evaluations: - \( \text{det}(2A) = 2^3 \times (-5) \) - \( \text{det}(A^2) = (-5)^2 \) - \( \text{det}(A^{-1}) = \frac{1}{-5} \) Completing these calculations will provide the evaluated determinants for each expression.
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