JEE Advance - Mathematics (2011 - Paper 2 Offline - No. 19)
Let M be a 3 $$\times$$ 3 matrix satisfying $$M\left[ {\matrix{
0 \cr
1 \cr
0 \cr
} } \right] = \left[ {\matrix{
{ - 1} \cr
2 \cr
3 \cr
} } \right]$$, $$M\left[ {\matrix{
1 \cr
{ - 1} \cr
0 \cr
} } \right] = \left[ {\matrix{
1 \cr
1 \cr
{ - 1} \cr
} } \right]$$ and $$M\left[ {\matrix{
1 \cr
1 \cr
1 \cr
} } \right] = \left[ {\matrix{
0 \cr
0 \cr
{12} \cr
} } \right]$$. Then the sum of the diagonal entries of M is ___________.
Answer
9
Explanation
Let $$M = \left[ {\matrix{ a & b & c \cr d & e & f \cr g & h & i \cr } } \right]$$
$$M = \left[ {\matrix{ 0 \cr 1 \cr 0 \cr } } \right] = \left[ {\matrix{ { - 1} \cr 2 \cr 3 \cr } } \right] \Rightarrow b = - 1,\,e = 2,\,h = 3$$
$$M = \left[ {\matrix{ 1 \cr { - 1} \cr 0 \cr } } \right] = \left[ {\matrix{ 1 \cr 1 \cr { - 1} \cr } } \right] \Rightarrow a = 0,\,d = 3,\,g = 2$$
$$M = \left[ {\matrix{ 1 \cr 1 \cr 1 \cr } } \right] = \left[ {\matrix{ 0 \cr 0 \cr {12} \cr } } \right] \Rightarrow g + h + i = 12 \Rightarrow i = 7$$
Hence, the sum of diagonal elements is 9.
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