Question

If A1, B1, C1 … are the cofactors of the elements a1, b1, c1... of the matrix

$\left[\begin{array}{ccc}a1& b1& c1\\ a2& b2& c2\\ a3& b3& c3\end{array}\right]$ then $\left|\begin{array}{cc}B2& C2\\ B3& C3\end{array}\right|=$

• A. a1 D
• B. a1a3D
• C. (a1 +a2)D
• D. None of these

Answer 1

The correct option is A

a1 D

Here as per the statement of the question B2, C2, B3 and C3 are the co factors of b2,c2 b3 and c3 respectively. So let’s calculate them first.

$B2=(-1{)}^{2+2}\left|\begin{array}{cc}a1& c1\\ a3& c3\end{array}\right|=(a1c3-c1a3)$

$C2=(-1{)}^{2+3}\left|\begin{array}{cc}a1& b1\\ a3& b3\end{array}\right|=-(a1b3-a3b1)$

$B3=(-1{)}^{3+2}\left|\begin{array}{cc}a1& c1\\ a2& c2\end{array}\right|=-(a1c2-a2c1)$

$C3=(-1{)}^{3+3}\left|\begin{array}{cc}a1& b1\\ a2& b2\end{array}\right|=(a1b2-a2b1)$

$\left|\begin{array}{cc}B2& C2\\ B3& C3\end{array}\right|=\left|\begin{array}{cc}a1c3-a3c1& -(a1b3-a3b1)\\ -(a1c2-a2c1)& a1b2-a2b1\end{array}\right|$

Solving this we will get

$a{1}^2(b2c3-b3c2)+a1b1(a3c2-c3a2)+a1c1(-a3b2+a2b3)+c1b1(a3a2-a2a3)$

$=a1(\underset{–}{a1(b2c3-b3c2)+b1(a3c2-c3a2)+c1(-a3b2+a2b3)}+0)$

Now if you look carefully, the underlined part is the value of the determinant D expanded along the first row

So, $\left|\begin{array}{cc}B2& C2\\ B3& C3\end{array}\right|=a1D$ which is the option (a)