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Chapter 3: Problem 5

Prove that \(E\) is an elementary matrix if and only if \(E^{t}\) is.

Short Answer

Expert verified
Proof: We analyzed each of the three types of elementary matrices (Type 1, Type 2, and Type 3) and found that for each type, if E is an elementary matrix, its transpose \(E^{t}\) is also an elementary matrix. Therefore, this holds for all elementary matrices, and we can conclude that E is an elementary matrix if and only if \(E^{t}\) is an elementary matrix.

Step by step solution

01

Define an elementary matrix

An elementary matrix is a matrix that can be obtained from the identity matrix by performing exactly one elementary row operation. There are three types of elementary matrices: 1. Type 1: Obtained by swapping two rows of the identity matrix 2. Type 2: Obtained by multiplying a row of the identity matrix by a non-zero constant 3. Type 3: Obtained by adding/subtracting a multiple of one row of the identity matrix to another row Now, let's check each type of elementary matrix and see if their transposes also form elementary matrices.
02

Analyze Type 1 elementary matrix and its transpose

A Type 1 elementary matrix is created by swapping two rows of the identity matrix. Let E be a Type 1 elementary matrix, and let \(E^{t}\) be its transpose. To obtain \(E^{t}\), we swap the corresponding columns of E. But since E was obtained from the identity matrix by swapping rows, swapping columns will result in another identity matrix with a pair of swapped columns. Thus, \(E^{t}\) is also an elementary matrix of Type 1.
03

Analyze Type 2 elementary matrix and its transpose

A Type 2 elementary matrix is created by multiplying a row of the identity matrix by a non-zero constant. Let E be a Type 2 elementary matrix, and let \(E^{t}\) be its transpose. To obtain \(E^{t}\), we replace the corresponding row of E with the same row multiplied by the non-zero constant. Since E was obtained from the identity matrix by multiplying a row by a non-zero constant, replacing the row with the multiplied row will result in another identity matrix with one row multiplied by the same non-zero constant. Thus, \(E^{t}\) is also an elementary matrix of Type 2.
04

Analyze Type 3 elementary matrix and its transpose

A Type 3 elementary matrix is created by adding/subtracting a multiple of one row of the identity matrix to another row. Let E be a Type 3 elementary matrix, and let \(E^{t}\) be its transpose. To obtain \(E^{t}\), we add/subtract the corresponding columns of E scaled by the same multiple. But since E was obtained from the identity matrix by adding/subtracting a multiple of one row to another row, adding/subtracting the scaled columns will result in another identity matrix with the addition/subtraction of the same multiple of one column to another column. Thus, \(E^{t}\) is an elementary matrix of Type 3.
05

Conclusion

In conclusion, we have proven that if E is an elementary matrix, then its transpose \(E^{t}\) is also an elementary matrix. This holds for all three types of elementary matrices. Therefore, E is an elementary matrix if and only if \(E^{t}\) is an elementary matrix.

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Most popular questions from this chapter

Let \(A\) be an \(m \times n\) matrix. Prove that there exists a sequence of elementary row operations of types 1 and 3 that transforms \(A\) into an upper triangular matrix.

Label the following statements as true or false. (a) An elementary matrix is always square. (b) The only entries of an elementary matrix are zeros and ones. (c) The \(n \times n\) identity matrix is an elementary matrix. (d) The product of two \(n \times n\) elementary matrices is an elementary matrix. (e) The inverse of an elementary matrix is an elementary matrix. (f) The sum of two \(n \times n\) elementary matrices is an elementary matrix. (g) The transpose of an elementary matrix is an elementary matrix. (h) If \(B\) is a matrix that can be obtained by performing an elementary row operation on a matrix \(A\), then \(B\) can also be obtained by performing an elementary column operation on \(A\). (i) If \(B\) is a matrix that can be obtained by performing an elementary row operation on a matrix \(A\), then \(A\) can be obtained by performing an elementary row operation on \(B\).

Find the inverse of each of the following \(n \times n\) matrices: (a) \(A\) has 1 's on the diagonal and superdiagonal (entries directly above the diagonal) and 0 's elsewhere (b) \(B\) has 1 's on and above the diagonal, and 0 's below the diagonal.

Determine whether each of the following systems is linear: (a) \(3 x-4 y+2 y z=8\) (b) \(\quad e x+3 y=\pi\) (c) \(2 x-3 y+k z=4\)

Prove Theorem 3.17: Let \(A\) be a square matrix. Then the following are equivalent: (a) \(A\) is invertible (nonsingular) (b) \(A\) is row equivalent to the identity matrix \(I\) (c) \(A\) is a product of elementary matrices.
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