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Chapter 3: Problem 6

Let \(A\) be an \(m \times n\) matrix. Prove that if \(B\) can be obtained from \(A\) by an elementary row [column] operation, then \(B^{t}\) can be obtained from \(A^{t}\) by the corresponding elementary column [row] operation.

Short Answer

Expert verified
We proved the statement by considering the three types of elementary row and column operations. For each type, we have shown that if matrix $B$ can be obtained from matrix $A$ by an elementary row operation, then the transpose of matrix $B$ is obtained from the transpose of matrix $A$ by the corresponding elementary column operation. In essence, this demonstrates that the transpose operation preserves the row and column operations of a matrix.

Step by step solution

01

Types of Elementary Row and Column Operations

There are three types of elementary row operations and their corresponding elementary column operations: 1. Interchange of Row i and Row j (or Column i and Column j) - This operation exchanges the positions of Row i and Row j (or Column i and Column j). 2. Multiplying elements in Row i (or Column i) by a nonzero scalar k. - This operation scales Row i (or Column i) by the nonzero scalar k. 3. Replacing Row i (or Column i) by the sum of Row i (or Column i) with k times Row j (or Column j), where k is any scalar. We will now show for each type of elementary row operation how we can apply a corresponding elementary column operation on the transpose of the matrix to achieve the same result.
02

Type 1: Interchange of Rows (Columns) in A and A^t

Suppose we perform an elementary row operation of type 1, say exchange of Row i and Row j in A to obtain matrix B. Now, if we take the transpose of A (denoted as A^t), this will mean that Row i and Row j in A converts to Column i and Column j in A^t. Therefore, if we perform the elementary column operation of type 1, i.e., exchange of Column i and Column j in A^t, it will result in B^t, as required.
03

Type 2: Multiplying Rows (Columns) in A and A^t

Suppose we perform an elementary row operation of type 2, say multiply Row i in A by a nonzero scalar k to obtain matrix B. Now, if we take the transpose of A (A^t), this would mean that Row i in A becomes Column i in A^t. Therefore, if we perform the elementary column operation of type 2, i.e., multiply Column i in A^t by the same nonzero scalar k, it will result in B^t, as required.
04

Type 3: Replacing Rows (Columns) in A and A^t

Suppose we perform an elementary row operation of type 3, say replace Row i in A with the sum of Row i and k times Row j in A to obtain matrix B. Now, if we take the transpose of A (A^t), this would mean that Row i in A becomes Column i in A^t and Row j in A becomes Column j in A^t. Therefore, if we perform the elementary column operation of type 3, i.e., replace Column i in A^t with the sum of Column i and k times Column j in A^t, it will result in B^t, as required. We have now shown that for each type of elementary row operation performed on A, we can perform the corresponding elementary column operation in A^t to obtain B^t. Hence, the statement is proven.

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Most popular questions from this chapter

For each of the following matrices, compute the rank and the inverse if it exists. (a) \(\left(\begin{array}{ll}1 & 2 \\ 1 & 1\end{array}\right)\) (b) \(\left(\begin{array}{ll}1 & 2 \\ 2 & 4\end{array}\right)\) (c) $\left(\begin{array}{rrr}1 & 2 & 1 \\ 1 & 3 & 4 \\ 2 & 3 & -1\end{array}\right)$ (d) $\left(\begin{array}{rrr}0 & -2 & 4 \\ 1 & 1 & -1 \\ 2 & 4 & -5\end{array}\right)$ (e) $\left(\begin{array}{rrr}1 & 2 & 1 \\ -1 & 1 & 2 \\ 1 & 0 & 1\end{array}\right)$ (f) $\left(\begin{array}{lll}1 & 2 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 1\end{array}\right)$ (g) $\left(\begin{array}{rrrr}1 & 2 & 1 & 0 \\ 2 & 5 & 5 & 1 \\ -2 & -3 & 0 & 3 \\ 3 & 4 & -2 & -3\end{array}\right)$ (h) $\left(\begin{array}{rrrr}1 & 0 & 1 & 1 \\ 1 & 1 & -1 & 2 \\ 2 & 0 & 1 & 0 \\ 0 & -1 & 1 & -3\end{array}\right)$

Let \(A\) be an \(m \times n\) matrix. Prove that if \(c\) is any nonzero scalar, then \(\operatorname{rank}(c A)=\operatorname{rank}(A)\).

Find the augmented matrix \(M\) and the coefficient matrix \(A\) of the following system: $$\begin{aligned} x+2 y-3 z &=4 \\\3 y-4 z+7 x &=5 \\\6 z+8 x-9 y &=1\end{aligned}$$

Let \(A\) be an \(m \times n\) matrix with rank \(m\) and \(B\) be an \(n \times p\) matrix with rank \(n\). Determine the rank of \(A B\). Justify your answer.

Show that every elementary matrix \(E\) is invertible, and its inverse is an elementary matrix.
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