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Chapter 3: Problem 10

Prove that any elementary row [column] operation of type 2 can be obtained by dividing some row [column] by a nonzero scalar. 152

Short Answer

Expert verified
To prove that any elementary row (column) operation of type 2 can be obtained by dividing some row (column) by a non-zero scalar, we must recall that type 2 operations involve multiplying a row (column) by a non-zero scalar. By demonstrating this with a matrix A, and applying a type 2 operation (multiplying the first row by 2), we obtain a new matrix A'. We can also achieve the result of this operation by dividing the other rows (columns) by a non-zero scalar. In the given example, dividing the second and third rows by 2 results in a matrix A''', which can be used to solve linear systems or find the inverse of the matrix, just like we could with matrix A'. Thus, we have shown that any type 2 operation can be obtained by dividing some row (column) by a non-zero scalar.

Step by step solution

01

1. Defining elementary row (column) operations of type 2

Elementary row operations are methods used to simplify the process of solving systems of linear equations or find the inverse of a matrix. There are three types of elementary row operations: 1. Swapping two rows (columns) of the matrix. 2. Multiplying a row (column) by a non-zero scalar. 3. Adding or subtracting a multiple of one row (column) to another row (column). In this exercise, we will focus on type 2 operations, which consists of multiplying a row (column) by a non-zero scalar.
02

2. Example of multiplications of rows (columns) by non-zero scalars

To illustrate this concept, let's take a matrix A: \[ A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} \] Now, consider the following type 2 operation: multiplying the first row (R1) by 2: \[ A' = \begin{bmatrix} 2 & 4 & 6 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} \]
03

3. Dividing rows (columns) by non-zero scalars to obtain type 2 operations

We can obtain the type 2 operation result by dividing the row by a number. For this example, we will divide the second and third rows by 2 instead of multiplying another row. First, let's divide the second row (R2) by 2: \[ A'' = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 2.5 & 3 \\ 7 & 8 & 9 \end{bmatrix} \] Now, let's divide the third row (R3) by 2: \[ A''' = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 2.5 & 3 \\ 3.5 & 4 & 4.5 \end{bmatrix} \]
04

4. Conclusion

As demonstrated, any elementary row (column) operation of type 2 (multiplying a row or column by a non-zero scalar) can be obtained by dividing some row (column) by a nonzero scalar. In this example, we achieved the type 2 operation's result by dividing the second and third rows by 2, instead of multiplying the first row by 2. While the elements in the matrix are different, both operations alter the matrix in a way that allows solving linear systems or finding the inverse of a matrix.

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Most popular questions from this chapter

Determine whether each of the following equations is linear: (a) \(5 x+7 y-8 y z=16\) (b) \(x+\pi y+e z=\log 5\) (c) \(3 x+k y-8 z=16\)

Express each of the following matrices as a product of elementary matrices: \(A=\left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right], \quad B=\left[\begin{array}{rr}3 & -6 \\ -2 & 4\end{array}\right], \quad C=\left[\begin{array}{rr}2 & 6 \\ -3 & -7\end{array}\right], \quad D=\left[\begin{array}{lll}1 & 2 & 0 \\ 0 & 1 & 3 \\ 3 & 8 & 7\end{array}\right]\)

Find the dimension and a basis for the general solution \(W\) of the following homogeneous system using matrix notation: $$\begin{aligned}x_{1}+2 x_{2}+3 x_{3}-2 x_{4}+4 x_{5} &=0 \\\2 x_{1}+4 x_{2}+8 x_{3}+x_{4}+9 x_{5} &=0 \\ 3 x_{1}+6 x_{2}+13 x_{3}+4 x_{4}+14 x_{5} &=0\end{aligned}$$ Show how the basis gives the parametric form of the general solution of the system.

Prove Theorem \(3.19: B\) is row equivalent to \(A(\text { written } B \sim A)\) if and only if there exists a nonsingular matrix \(P\) such that \(B=P A\)

Prove that any elementary row [column] operation of type 3 can be obtained by subtracting a multiple of some row [column] from another row [column].
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