Question

Using properties of determinants, the value of determinant
$\left|\begin{array}{ccc}18& 40& 89\\ 40& 89& 198\\ 89& 198& 440\end{array}\right|$ is obtained as:

• A. $3$
• B. $1$
• C. $-1$
• D. $0$

Answer 1

The correct option is C $-1$

Let $D=\left|\begin{array}{ccc}18& 40& 89\\ 40& 89& 198\\ 89& 198& 440\end{array}\right|$
Applying $R_2\to R_2-2R_1$ and $R_3\to R_3-5R_1$
$\Rightarrow D=\left|\begin{array}{ccc}18& 40& 89\\ 4& 9& 20\\ -1& -2& -5\end{array}\right|$
Applying $C_3\to C_3-2C_2$
$D=\left|\begin{array}{ccc}18& 40& 9\\ 4& 9& 2\\ -1& -2& -1\end{array}\right|$
Now applying $C_1\to C_1-2C_3$ to get zeros in $C_1$
$D=\left|\begin{array}{ccc}0& 40& 9\\ 0& 9& 2\\ 1& -2& -1\end{array}\right|=1\left|\begin{array}{cc}40& 9\\ 9& 2\end{array}\right|\Rightarrow D=80-81=-1$