Question

Let A be an $n\times n$ matrix with rank $r(0<r<n$). Then AX=0 has p independent solutions, where p is

• A. r
• B. n
• C. n - r
• D. n + r

Answer 1

The correct option is C n - r

The system AX=0 has $(n-r$) linearly independent solutions; where 'n' is number of unknowns and 'r' is rank of 'A'
For homogeneous system AX=0
$p=(n-r$)
Nullity = Number of independent solution
= Number of columns - rank
So $p=n-r$