Question

The value of determinant $\left|\begin{array}{ccc}2& 7& 65\\ 3& 8& 75\\ 5& 9& 86\end{array}\right|equals$.

Answer 1

We know that to solve a determinant, we apply the following formula:

$\left|\begin{array}{ccc}{a}_1& a_2& a_3\\ b_1& b_2& b_3\\ c_1& c_2& c_3\end{array}\right|=a_1[(b_2\times c_3)-(c_2\times b_3)]-a_2[(b_1\times c_3)-(c_1\times b_3)]+a_3[(b_1\times c_2)-(c_1\times b_2)]$

Now, consider the given determinant and solve as follows:

$\left|\begin{array}{ccc}2& 7& 65\\ 3& 8& 75\\ 5& 9& 86\end{array}\right|=2[(8\times 86)-(9\times 75)]-7[(3\times 86)-(5\times 75)]+65[(3\times 9)-(5\times 8)]=2(688-675)-7(258-375)+65(27-40)=(2\times 13)-(7\times -117)+(65\times -13)=26+819-845=845-845=0$

Hence, $\left|\begin{array}{ccc}2& 7& 65\\ 3& 8& 75\\ 5& 9& 86\end{array}\right|=0$