Math Problem Statement
Calculate the determinants of matrices A and B and their transformations, given specific matrix operations.
Solution
The image contains a matrix problem, where you are asked to calculate several determinants. I'll help you understand the questions and provide explanations for each part.
Given matrices:
1 & -3 & -4 & 1 \\ -3 & 0 & 1 & 2 \\ 2 & -1 & 1 & 1 \end{pmatrix}, \quad B = \begin{pmatrix} 0 & -4 & -1 \\ 0 & 4 & 2 \\ 1 & 0 & -2 \end{pmatrix}$$ You're required to calculate the following: 1. **det(A) \* det(B)** 2. **det(4A)** 3. **det(A^{-1})** 4. **det(B^{-1})** 5. **det(A^T)** Let's go through them step by step. ### a. $$\text{det}(A) \times \text{det}(B)$$ - $$\text{det}(A)$$ is the determinant of matrix $$A$$. - $$\text{det}(B)$$ is the determinant of matrix $$B$$. Using the properties of determinants: $$\text{det}(A) = 0, \quad \text{det}(B) = 0$$ Since the product of two determinants involves multiplying their values: $$\text{det}(A) \times \text{det}(B) = 0 \times 0 = 0$$ ### b. $$\text{det}(4A)$$ This uses the property: $$\text{det}(kA) = k^n \times \text{det}(A)$$ where $$k = 4$$ and $$n$$ is the order of the matrix $$A$$. Since $$A$$ is a 3x3 matrix, $$n = 3$$: $$\text{det}(4A) = 4^3 \times \text{det}(A) = 64 \times (-23) = -1472$$ ### c. $$\text{det}(A^{-1})$$ Using the property: $$\text{det}(A^{-1}) = \frac{1}{\text{det}(A)}$$ Given that $$\text{det}(A) = -23$$: $$\text{det}(A^{-1}) = \frac{1}{-23} = -\frac{1}{23}$$ ### d. $$\text{det}(B^{-1})$$ The inverse of a matrix exists only if its determinant is non-zero. Since $$\text{det}(B) = 0$$, matrix $$B$$ is singular, meaning it has no inverse: $$\text{det}(B^{-1}) = \text{none}$$ ### e. $$\text{det}(A^T)$$ The determinant of the transpose of a matrix is the same as the determinant of the original matrix: $$\text{det}(A^T) = \text{det}(A) = -23$$ --- Would you like further details or have any questions? Here are 5 related questions to consider: 1. How is the determinant of a 3x3 matrix computed step by step? 2. Why does a zero determinant imply that a matrix has no inverse? 3. Can you explain how the transpose of a matrix affects its determinant? 4. How does the scalar multiplication of a matrix affect its determinant? 5. What are the implications of a matrix being singular in solving systems of equations? **Tip**: When computing determinants for larger matrices, break down the matrix into smaller 2x2 matrices using cofactor expansion.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Determinants
Matrix Inverse
Matrix Transpose
Scalar Multiplication
Formulas
det(kA) = k^n * det(A) where k is a scalar and n is the matrix order
det(A * B) = det(A) * det(B)
det(A^T) = det(A)
det(A^-1) = 1 / det(A)
Theorems
Determinant of product of matrices
Determinant of inverse matrix
Determinant of scalar multiplication
Determinant of transpose matrix
Suitable Grade Level
Undergraduate