Question

Arrange the given matrices in ascending order of their ranks.

A) $\left[\begin{array}{ccc}1& 4& -1\\ 2& 3& 0\\ 0& 1& 2\end{array}\right]$ B) $\left[\begin{array}{ccc}1& 1& 1\\ 1& 1& 1\\ 1& 1& 1\end{array}\right]$ C) $\left[\begin{array}{ccc}1& 2& 3\\ 2& 3& 4\\ 0& 1& 2\end{array}\right]$

• A. A,B,C
• B. A,C,B
• C. B,C,A
• D. C,A,B

Answer 1

Answer: (C)

B,C,A

$B,C,A$

A

$\left(\begin{array}{ccc}1& 4& -1\\ 2& 3& 0\\ 0& 1& 2\end{array}\right)$

$R_1\leftrightarrow R_2$

$R_2\leftarrow R_2-\frac{1}{2}\cdot R_1$

$\left(\begin{array}{ccc}2& 3& 0\\ 0& \frac{5}{2}& -1\\ 0& 1& 2\end{array}\right)$

$R_3\leftarrow R_3-\frac{2}{5}\cdot R_2$

$\left(\begin{array}{ccc}2& 3& 0\\ 0& \frac{5}{2}& -1\\ 0& 0& \frac{12}{5}\end{array}\right)$

$R_3\leftarrow \frac{5}{12}\cdot R_3$

$R_2\leftarrow R_2+1\cdot R_3$

$\left(\begin{array}{ccc}2& 3& 0\\ 0& \frac{5}{2}& 0\\ 0& 0& 1\end{array}\right)$

$R_2\leftarrow \frac{2}{5}\cdot R_2$

$R_1\leftarrow R_1-3\cdot R_2$

$\left(\begin{array}{ccc}2& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)$

$R_1\leftarrow \frac{1}{2}\cdot R_1$

$\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)$

Rank $=3$

B

$\left(\begin{array}{ccc}1& 1& 1\\ 1& 1& 1\\ 1& 1& 1\end{array}\right)$

$R_2\leftarrow R_2-1\cdot R_1$

$R_3\leftarrow R_3-1\cdot R_1$

$\left(\begin{array}{ccc}1& 1& 1\\ 0& 0& 0\\ 0& 0& 0\end{array}\right)$

Rank $=1$

C

$\left(\begin{array}{ccc}1& 2& 3\\ 2& 3& 4\\ 0& 1& 2\end{array}\right)$

$R_1\leftrightarrow R_2$

$R_2\leftarrow R_2-\frac{1}{2}\cdot R_1$

$\left(\begin{array}{ccc}2& 3& 4\\ 0& \frac{1}{2}& 1\\ 0& 1& 2\end{array}\right)$

$R_2\leftrightarrow R_3$

$R_3\leftarrow R_3-\frac{1}{2}\cdot R_2$

$\left(\begin{array}{ccc}2& 3& 4\\ 0& 1& 2\\ 0& 0& 0\end{array}\right)$

Rank $=2$