Question

Are there values of r and s for which [1,0,0,0,r-2,2,0,s-1,r+2,0,0,3] has rank 1 or 2? If so, find those values

Answer 1

$\begin{bmatrix} 1 & 0 &0 \\ 0 & r-2 &2\\ 0 &s-1 &r+2\\ 0& 0 &3\\ \end{bmatrix}$

The matrix has rank 2

Therefore, any two columns or rows must be zero.

If r=2 and s=1;

$\begin{bmatrix} 1 & 0 &0 \\ 0 & 0&2\\ 0 &0 &4\\ 0& 0 &3\\ \end{bmatrix}$

Converting rows 2, 3 and 4 to unity

$\begin{bmatrix} 1 & 0 &0 \\ 0 & 0&1\\ 0 &0 &1\\ 0& 0 &1\\ \end{bmatrix}$

R₂-R₃ and R₂-R₄;

$\begin{bmatrix} 1 & 0 &0 \\ 0 & 0&1\\ 0 &0 &0\\ 0& 0 &0\\ \end{bmatrix}$

So,

Rank of matrix=2

r=2

s=1