If $A=\left[a_{i j}\right], 1 \leq i, j \leq n$ with $\mathrm{n} \geq 2$ and $a_{i j}=i+j$ is a matrix, then…
If $A=\left[a_{i j}\right], 1 \leq i, j \leq n$ with $\mathrm{n} \geq 2$ and $a_{i j}=i+j$ is a matrix, then the rank of $A$ is
- 0
- 1
- 2
- 4
Solution
Given that $n \geq 2$ and $a_{\mathrm{ij}}=i+j$
Case-1: Let $n=2$
$A=\left[\begin{array}{ll}
2 & 3 \\
3 & 4
\end{array}\right] \Rightarrow|A|=\left|\begin{array}{ll}
2 & 3 \\
3 & 4
\end{array}\right|=8-1=-1 \neq 0$
Case-2: Let $n=3$
$A=\left[\begin{array}{lll}
2 & 3 & 4 \\
3 & 4 & 5 \\
4 & 5 & 6
\end{array}\right] \Rightarrow|A|=\left|\begin{array}{lll}
2 & 3 & 4 \\
3 & 4 & 5 \\
4 & 5 & 6
\end{array}\right|=0$
Hence rank $=2$
Asked in: AP EAMCET 2024 (20 May Shift 2)
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