JEE Advanced 2011 Paper 2Question 1
Let $M$ be a $3\times3$ matrix satisfying $$M\begin{bmatrix}0\\1\\0\end{bmatrix}=\begin{bmatrix}-1\\2\\3\end{bmatrix},\ M\begin{bmatrix}1\\-1\\0\end{bmatrix}=\begin{bmatrix}1\\1\\-1\end{bmatrix},\text{ and }M\begin{bmatrix}1\\1\\1\end{bmatrix}=\begin{bmatrix}0\\0\\12\end{bmatrix}.$$ Then the sum of the diagonal entries of $M$ is
Matrices
Answer$9$
Solution
Column 2 of $M$ is $(-1,2,3)^T$. $M(1,-1,0)^T=C_1-C_2=(1,1,-1)^T$ gives $C_1=(0,3,2)^T$. $C_1+C_2+C_3=(0,0,12)^T$ gives $C_3=(1,-5,7)^T$. Trace $=0+2+7=9$.