Question

Consider an n×mmatrix A. Can you transform rref(A)into Aby a sequence of elementary row operations? See Exercise 27.

Short answer

Yes, for transforming the matrix A into rref we can do the sequence of operations.

Step-by-step solution

Row operations

In matrix there are three types of row operations for transforming the matrix of the form n×m:

1. Swapping the rows: In this operation, we can swap any row with any other row.
2. Constant multiplications with row: In this operation, we can multiply any non-zero scalar with any row.
3. Subtracting or adding the row: In this operation, we can subtract or add another row.

Example for explanation

Suppose a matrix A of order 2×2,$A=\left[\begin{array}{cc}1& 3\\ 3& 6\end{array}\right]$

We will use the sequence of row operations for transforming the equation to rref.

$$\begin{gathered} A=\left[\begin{array}{cc}1& 3\\ 3& 6\end{array}\right] \\ {R}_2\to R_2-3R_1 \\ ~\left[\begin{array}{cc}1& 3\\ 0& -3\end{array}\right] \\ {R}_2\to \frac{1}{-3}{R}_2 \\ ~\left[\begin{array}{cc}1& 3\\ 0& 1\end{array}\right] \\ {R}_1\to R_1-3R_2 \\ ~\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right] \end{gathered}$$

Yes, for transforming the matrix A into rref we can do the sequence of operations.