Question

Let $A=\left[\begin{array}{ccc}-1& -2& -3\\ 3& 4& 5\\ 4& 5& 6\end{array}\right],B=\left[\begin{array}{cc}1& -2\\ -1& 2\end{array}\right]$ and $C=\left[\begin{array}{ccc}2& 0& 0\\ 0& 2& 0\\ 0& 0& 2\end{array}\right]$. If $a,b$ and $c$ respectively denote the ranks of $A,B$ and $C$ then the correct order of these numbers is

• A. $a<b<c$
• B. $c<b<a$
• C. $b<a<c$
• D. $a<c<b$

Answer 1

Answer: (C)

$b<a<c$

$C=\left[\begin{array}{ccc}2& 0& 0\\ 0& 2& 0\\ 0& 0& 2\end{array}\right]$

It has three linear independent rows.

Rank of $C=3$

$B=\left[\begin{array}{cc}1& -2\\ -1& 2\end{array}\right]$

$=\left[\begin{array}{cc}1& -2\\ 0& 0\end{array}\right]$

Rank of $B=1$

$A=\left[\begin{array}{ccc}-1& -2& -3\\ 3& 4& 5\\ 4& 5& 6\end{array}\right]$

rank of A =2

$b<a<c$