Question

Find the rank of the matrix
$A=\left[\begin{array}{cccc}{1}^2& 2^2& 3^2& 4^2\\ 2^2& 3^2& 4^2& 5^2\\ 3^2& 4^2& 5^2& 6^2\\ 4^2& 5^2& 6^2& 7^2\end{array}\right]$

• A. 2
• B. 3
• C. 4
• D. None of these

Answer 1

The correct option is B 3

$A=\left[\begin{array}{cccc}{1}^2& 2^2& 3^2& 4^2\\ 2^2& 3^2& 4^2& 5^2\\ 3^2& 4^2& 5^2& 6^2\\ 4^2& 5^2& 6^2& 7^2\end{array}\right]$
$|A|=\left|\begin{array}{cccc}1& 4& 9& 16\\ 4& 9& 16& 25\\ 9& 16& 25& 36\\ 16& 25& 36& 49\end{array}\right|$
$C_4\to C_4-C_3,C_3\to C_3-C_2{,C}_2\to C_2-C_1$
$\left|\begin{array}{cccc}1& 3& 5& 7\\ 4& 5& 7& 9\\ 9& 7& 9& 11\\ 16& 9& 11& 13\end{array}\right|$
$C_4\to C_4-C_3,C_3\to C_3-C_2{,C}_2\to C_2-C_1$
$\left|\begin{array}{cccc}1& 2& 2& 2\\ 4& 1& 2& 2\\ 9& -2& 2& 2\\ 16& -7& 2& 2\end{array}\right|$
$R_2\to R_2-R_1,R_3\to R_3-R_1{,R}_4\to R_4-R_1$
$\left|\begin{array}{cccc}1& 2& 2& 2\\ 3& -1& 0& 0\\ 8& -4& 0& 0\\ 15& -9& 0& 0\end{array}\right|$
Expanding along last column, we get $|A|=0$
So, rank of A $\ne 4$
Now, we will look for minors of order 3,
Here, $\left|\begin{array}{ccc}1& 2& 2\\ 3& -1& 0\\ 8& -4& 0\end{array}\right|\ne 0$
Hence, rank of matrix =3