Question

Which of the following has/have value equal to zero?

• A. $\left|\begin{array}{ccc}8& 2& 7\\ 12& 3& 5\\ 16& 4& 3\end{array}\right|$
• B. $\left|\begin{array}{ccc}1/a& a^2& bc\\ 1/b& b^2& ac\\ 1/c& c^2& ab\end{array}\right|$
• C. $\left|\begin{array}{ccc}a+b& 2a+b& 3a+b\\ 2a+b& 3a+b& 4a+b\\ 4a+b& 5a+b& 6a+b\end{array}\right|$
• D. $\left|\begin{array}{ccc}2& 43& 6\\ 7& 35& 4\\ 3& 17& 2\end{array}\right|$

Answer 1

Answer: (A), (B), (C)

The correct options are
A $\left|\begin{array}{ccc}8& 2& 7\\ 12& 3& 5\\ 16& 4& 3\end{array}\right|$
B $\left|\begin{array}{ccc}1/a& a^2& bc\\ 1/b& b^2& ac\\ 1/c& c^2& ab\end{array}\right|$
C $\left|\begin{array}{ccc}a+b& 2a+b& 3a+b\\ 2a+b& 3a+b& 4a+b\\ 4a+b& 5a+b& 6a+b\end{array}\right|$

(A) $\left|\begin{array}{ccc}8& 2& 7\\ 12& 3& 5\\ 16& 4& 3\end{array}\right|=8(3\times 3-4\times 5)-2(12\times 3-16\times 5)+7(12\times 4-3\times 16)$

$=8(9-20)-2(36-80)+7(48-48)=-88+88+0=0.$

(B) $\left|\begin{array}{ccc}\frac{1}{a}& a^2& bc\\ \frac{1}{b}& b^2& ac\\ \frac{1}{c}& c^2& ab\end{array}\right|=\frac{1}{a}(b^2\times ab-c^2\times ac)-a^2(\frac{1}{b}\times ab-\frac{1}{c}\times ac)+bc(\frac{1}{b}\times c^2-\frac{1}{c}\times b^2)$

$=b^3-c^3-a^3+a^3+c^3-b^3=0$

(C) $\left|\begin{array}{ccc}a+b& 2a+b& 3a+b\\ 2a+b& 3a+b& 4a+b\\ 4a+b& 5a+b& 6a+b\end{array}\right|=\left|\begin{array}{ccc}a& 2a& 3a\\ 2a& 3a& 4a\\ 4a& 5a& 6a\end{array}\right|+\left|\begin{array}{ccc}b& b& b\\ b& b& b\\ b& b& b\end{array}\right|$

$=\left(a\right)\left(a\right)\left(a\right)\left|\begin{array}{ccc}1& 2& 3\\ 2& 3& 4\\ 4& 5& 6\end{array}\right|+0$

$=a^3(1(3\times 6-5\times 4)-2(2\times 6-4\times 4)+3(2\times 5-3\times 4))$

$=a^3((18-20)-2(12-16)+3(10-12))=a^3(-2+8-6)=0$

(D) $\left|\begin{array}{ccc}2& 43& 6\\ 7& 35& 4\\ 3& 17& 2\end{array}\right|=2(35\times 2-4\times 17)-43(7\times 2-3\times 4)+6(7\times 17-3\times 35)$

$=2(70-68)-43(14-12)+6(119-105)=4-86+84=2.$