Question-31

row echelon form

Choose the correct set of options.

If two matrices \(A\) and \(B\) have the same reduced row echelon form, then \(A\) must be equal to \(B\).
If two matrices \(A\) and \(B\) have the same row echelon form, then \(A\) must be equal to \(B\).
The reduced row echelon form of a diagonal matrix with non-zero diagonal entries must be the identity matrix.
The reduced row echelon form of a non-zero scalar matrix must be the identity matrix.

Answer

If two matrices \(A\) and \(B\) have the same reduced row echelon form, then \(A\) must be equal to \(B\).
If two matrices \(A\) and \(B\) have the same row echelon form, then \(A\) must be equal to \(B\).
The reduced row echelon form of a diagonal matrix with non-zero diagonal entries must be the identity matrix.
The reduced row echelon form of a non-zero scalar matrix must be the identity matrix.

Solution

Option-(a) is incorrect. As a counter example, consider any two invertible matrices of the same order that are distinct from each other. Both have the same reduced row echelon form, namely the identity matrix.
Option-(b) is incorrect. An argument similar to the previous bullet point can be applied here.
Option-(c) is correct. In fact, as stated in the first point, any invertible matrix has the identity matrix as its reduced row echelon form. Since a diagonal matrix with non-zero diagonal entries is invertible, the result follows.
Option-(d) is correct. A non-zero scalar matrix is an example of a diagonal matrix with non-zero diagonal entries.