Find All 3 by 3 Reduced Row Echelon Form Matrices of Rank 1 and 2

Problem 646

(a) Find all $3 \times 3$ matrices which are in reduced row echelon form and have rank 1.

(b) Find all such matrices with rank 2.

Solution.

(a) Find all $3 \times 3$ matrices which are in reduced row echelon form and have rank 1.

First we look at the rank 1 case.
For a $3 \times 3$ matrix in reduced row echelon form to have rank 1, it must have 2 rows which are all 0s. The non-zero row must be the first row, and it must have a leading 1.

These conditions imply that the matrix must be of one of the following forms:
$\begin{bmatrix} 1 & a & b \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} , \quad \begin{bmatrix} 0 & 1 & c \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} , \mbox{ or } \begin{bmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}.$

(b) Find all such matrices with rank 2.

For a rank 2 $3 \times 3$ matrix in reduced row echelon form, there must be one row, the bottom one, which has only 0s.

Thus we need two leading 1s in distinct columns, and every other term in the same column with a leading 1 must be 0. The possibilities are:
$\begin{bmatrix} 1 & 0 & a \\ 0 & 1 & b \\ 0 & 0 & 0 \end{bmatrix} , \quad \begin{bmatrix} 1 & a & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{bmatrix} , \mbox{ or } \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{bmatrix}.$

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