Let $A=\left[\begin{array}{rrr}-1 & -2 & -3 \ 3 & 4 & 5 \ 4 & 5 & 6\end{array} ight], B=\left[\begin{array}{cr}1 & -2 \ -1 & 2\end{array} ight] \quad$ and $C=\left[\begin{array}{lll}2 & 0 & 0 \ 0 & 2 & 0 \ 0 & 0 & 2\end{array} ight]$, if $a, b$ and $c$ respectively, denote the ranks of $A,…

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Let $A=\left[\begin{array}{rrr}-1 & -2 & -3 \ 3 & 4 & 5 \ 4 & 5 & 6\end{array}ight], B=\left[\begin{array}{cr}1 & -2 \ -1 & 2\end{array}ight] \quad$ and $C=\left[\begin{array}{lll}2 & 0 & 0 \ 0 & 2 & 0 \ 0 & 0 & 2\end{array}ight]$, if $a, b$ and $c$ respectively, denote the ranks of $A, B$ and $C$, then the correct order of these number is

Aicharya · Accepted Answer

Given, $A=\left[\begin{array}{rrr}-1 & -2 & -3 \ 3 & 4 & 5 \ 4 & 5 & 6\end{array}ight]$ $\begin{aligned} 	herefore|A| & =-1(24-25)+2(18-20)-3(15-16) \ & =1-4+3=0\end{aligned}$ Now, $\left|\begin{array}{ll}4 & 5 \ 5 & 6\end{array}ight|=24-25=-1 
eq 0$ $	herefore$ Rank of $A=a=2$ $B=\left[\begin{array}{rr}1 & -2 \ -1 & 2\end{array}ight]$ $	herefore \quad|B|=2-2=0$ $	herefore$ Rank of $B=b=1$ and $\quad C=\left[\begin{array}{ccc}2 & 0 & 0 \ 0 & 2 & 0 \ 0 & 0 & 2\end{array}ight]$ $|C|=2(4-0)=8 
eq 0$ $	herefore$ Rank of $C=3, c=3$ $b < a < c \quad(\because 1 < 2 < 3)$

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

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