If $\det \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} = -15$ then $\det \begin{bmatrix} d & e & f \\ a & b & c \\ g & h & i \end{bmatrix} =$ [Blank space for answer]. ### Explanation:This problem involves calculating the determinants of 3x3 matrices. The given determinant is $-15$, and you are asked to determine the determinant of a second matrix where two rows have been swapped. In properties of determinants, swapping two rows changes the sign of the determinant. Hence, based on the given determinant, you can calculate the determinant of the second matrix by changing the sign of the first determinant.

Given that detabcdefghi=−15 We have to find the detdefabcghi Use the determinant property:- If any…

Answered: e f] b c If det d e f a 15 then det a b…

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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If $\det \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} = -15$ then $\det \begin{bmatrix} d & e & f \\ a & b & c \\ g & h & i \end{bmatrix} =$ [Blank space for answer].

### Explanation:
This problem involves calculating the determinants of 3x3 matrices. The given determinant is $-15$, and you are asked to determine the determinant of a second matrix where two rows have been swapped. In properties of determinants, swapping two rows changes the sign of the determinant. Hence, based on the given determinant, you can calculate the determinant of the second matrix by changing the sign of the first determinant.

Expert Solution

Step 1

Given that $\det [\begin{matrix} a & b & c \\ d & e & f \\ g & h & i \end{matrix}] = - 15$

We have to find the $\det [\begin{matrix} d & e & f \\ a & b & c \\ g & h & i \end{matrix}]$

Use the determinant property:- If any two row (or two column) of a determinant are interchanged the value of the determinant is multiplied by $- 1$ .

$\det [\begin{matrix} d & e & f \\ a & b & c \\ g & h & i \end{matrix}] = (- 1) \cdot \det [\begin{matrix} a & b & c \\ d & e & f \\ g & h & i \end{matrix}]$ ........(1)

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