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Chapter 4: Problem 29

Prove that if \(E\) is an elementary matrix, then \(\operatorname{det}\left(E^{t}\right)=\operatorname{det}(E) .\) Visit goo.gl/6ZoU5Z for a solution.

Short Answer

Expert verified
In summary, we proved that for any elementary matrix E, the determinant of its transpose (E^t) is equal to the determinant of E itself. We identified the three types of elementary matrices: row swapping (P), row addition (L), and row scaling (D). By examining each type separately, we showed that det(E) = det(E^t) for all three cases: 1. For row swapping (P), both det(P) and det(P^t) equal -1. 2. For row addition (L), both det(L) and det(L^t) equal 1. 3. For row scaling (D), both det(D) and det(D^t) equal the scalar used to scale a row. Hence, we have confirmed that if E is an elementary matrix, then det(E^t) = det(E).

Step by step solution

01

Identify Types of Elementary Matrices

There are three types of elementary matrices: 1. Type 1: Row Swapping (P) 2. Type 2: Row Addition (L) 3. Type 3: Row Scaling (D)
02

Step 2a: Prove for Row swapping

For a row swapping matrix P, we need to show that det(P) = det(P^t). The determinant of a row swapping matrix is -1. The transpose of a row swapping matrix is the same matrix (since swapping two rows and then taking the transpose is the same as taking the transpose and then swapping the corresponding columns). So, det(P^t) = det(P) = -1.
03

Step 2b: Prove for Row Addition

For a row addition matrix L, we need to show that det(L) = det(L^t). The determinant of a row addition matrix is 1, as adding one row to another doesn't change the determinant. The transpose of a row addition matrix L is still a row addition matrix (though it might represent adding columns instead of rows, the result remains the same). So, det(L^t) = det(L) = 1.
04

Step 2c: Prove for Row scaling

For a row scaling matrix D, we need to show that det(D) = det(D^t). The determinant of a row scaling matrix is the scalar which is used to scale a row. The transpose of a row scaling matrix is the same matrix D (since scaling a row and then taking the transpose is the same as taking the transpose and then scaling the corresponding column). So, det(D^t) = det(D).
05

Conclusion

We have shown that for every type of elementary matrix E (row swapping, row addition, and row scaling), det(E) = det(E^t). Therefore, we can conclude that for any elementary matrix E, det(E) = det(E^t).

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