Question

Suppose matrix Ais transformed into matrix Bby sequence of an elementary row operation. Is there an elementary row operation that transforms Binto A? Explain.

Short answer

Yes, for transforming the matrix B into A we can do the sequence inverse of operation that we have used for transforming the matrix A into B.

Step-by-step solution

Matrix transformation

If we have transformed any matrix A into B by using the sequence of row operations. Then we can transform B into A by using the sequence inverse of operations that we have used for transforming the matrix A into B.

Example of transformation

Suppose a matrix A and B be$A=\left[\begin{array}{cc}1& 3\\ 3& 6\end{array}\right]$and$B=\left[\begin{array}{cc}1& 3\\ 0& 1\end{array}\right]$

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

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

Now for transforming B into A we will reverse the operations like.

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

Now proceeding further

$$\begin{gathered} R_2\to R_2+3R_1 \\ ~\left[\begin{array}{cc}1& 3\\ 3& 6\end{array}\right]=A \end{gathered}$$

Hence, if we have transformed any matrix A into B by using the sequence of row operations. Then we can transform B into A by using the sequence inverse of operations that we have used for transforming the matrix A into B.