Math Problem Statement
State which property of determinants is illustrated in this equation. StartAbsoluteValue Start 3 By 3 Matrix 1st Row 1st Column 4 2nd Column negative 1 3rd Column 2 2nd Row 1st Column 8 2nd Column 9 3rd Column 7 3rd Row 1st Column negative 4 2nd Column negative 2 3rd Column 0 EndMatrix EndAbsoluteValue equals StartAbsoluteValue Start 3 By 3 Matrix 1st Row 1st Column 4 2nd Column negative 1 3rd Column 2 2nd Row 1st Column 0 2nd Column 11 3rd Column 3 3rd Row 1st Column negative 4 2nd Column negative 2 3rd Column 0 EndMatrix EndAbsoluteValue
Question content area bottom Part 1 Choose the correct answer below. A. If A and B are square matrices, then det AB equals left parenthesis det Upper A right parenthesis left parenthesis det Upper B right parenthesis. B. If two rows of A are interchanged to produce B, then det Upper B equals negative det Upper A. C. If a multiple of one row of A is added to another row to produce matrix B, then det Upper B equals det Upper A. D. If one row of A is multiplied by k to produce B, then det Upper B equals k times det Upper A.
Solution
The equation given shows that the determinant of the matrix does not change, even though the second row has undergone a transformation. Specifically, a multiple of one row of matrix A has been added to another row to produce matrix B.
This corresponds to Option C:
If a multiple of one row of A is added to another row to produce matrix B, then det(B) = det(A).
This is because adding a multiple of one row to another row is a determinant-preserving operation. Hence, the determinant of the new matrix (B) is the same as the determinant of the original matrix (A).
Would you like further details or have any questions?
Here are 5 related questions to expand on this:
- What is the determinant of a 3x3 matrix, and how is it calculated?
- How does the determinant change if two rows of a matrix are interchanged?
- What are the properties of matrix determinants under row operations?
- Why does adding a multiple of one row to another not affect the determinant?
- Can the determinant help in determining if a matrix is invertible?
Tip: The determinant can provide insights into the properties of a matrix, such as whether it is singular (non-invertible) or not.
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Determinants
Matrix Row Operations
Formulas
Det(A) = Det(B) if a multiple of one row of A is added to another row to produce B
Theorems
Determinant Preservation under Row Operations
Suitable Grade Level
Grades 11-12