Question-7

Consider the three elementary row operations:

Type-1: Interchanging two rows.

Type-2: Multiplying a row by a non-zero constant.

Type-3: Adding a scalar multiple of a row to another row.

For each statement below, prove it if it is correct and provide a counterexample if it isn’t.

  1. If a matrix \(A\) can be obtained from \(B\) by a finite number of row operations, then \(B\) can be obtained from \(A\) by a finite number of row operations.

  2. The reduced row echelon form of a matrix cannot be the identity matrix.

  3. An upper triangular matrix with all diagonal elements equal to \(1\) is in row echelon form.

  4. The identity matrix is in reduced row echelon form.

  5. The reduced row echelon form of a scalar matrix (other than identity) can be obtained by applying only operations of type 1.

  6. The reduced row echelon form of a diagonal matrix (other than identity) can be obtained by applying only operations of type-2.

(1) All row operations are reversible. Let us call the matrix \(A_{1}\) before the operation and let \(A_{2}\) be the matrix after performing the operation.

Type-1: If \(R_{1}\) and \(R_{2}\) are interchanged in \(A_{1}\), we can perform this operation on \(A_{2}\) to get back the original matrix \(A_{1}\).

Type-2: If \(R_{1}\) of \(A_{1}\) scaled by a non-zero constant \(c\), we can scale \(R_{1}\) of \(A_{2}\) by \(\frac{1}{c}\) to get back the original matrix \(A_{1}\).

Type-3: If \(R_{1}\) of \(A_{1}\) is replaced by \(R_{1}+R_{2}\), we can replace \(R_{1}\) of \(A_{2}\) by \(R_{1}-R_{2}\) to get back the original matrix \(A_{1}\).

(2) This is incorrect. The RREF of a matrix can be the identity matrix. The identity matrix is itself a trivial example. Every invertible matrix has the identity matrix as its RREF.

(3) This is true. As a concrete example, consider a \(3\times3\) matrix. All the entries not filled below can take arbitrary values:

\[ \begin{bmatrix}1\\ 0 & 1\\ 0 & 0 & 1 \end{bmatrix} \]

(4) Yes, the identity matrix is indeed in reduced row echelon form.

(5) A non-zero scalar matrix is of the form given below (\(c\neq0\)):

\[ cI=\begin{bmatrix}c & 0 & 0\\ 0 & c & 0\\ 0 & 0 & c \end{bmatrix} \]

To get its RREF, we need type-2 operation of scaling each row by the constant \(\frac{1}{c}\). Type-1 operation is going to be of no use here.

(6) To reduce a diagonal matrix to its RREF, we may need both type-1 and type-2 operations. This is especially the case if there are any zero entries on the diagonal. For instance:

\[ \begin{bmatrix}1 & 0 & 0\\ 0 & 0 & 0\\ 0 & 0 & 1 \end{bmatrix} \]

We need to perform \(R_{2}\leftrightarrow R_{3}\) to convert the above matrix into its RREF.