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Question
Let M be a 3×3 matrix satisfying
M⎡⎢⎣010⎤⎥⎦=⎡⎢⎣−123⎤⎥⎦,M⎡⎢⎣1−10⎤⎥⎦=⎡⎢⎣11−1⎤⎥⎦, and M⎡⎢⎣111⎤⎥⎦=⎡⎢⎣0012⎤⎥⎦.
Then the sum of the diagonal entries of M is
Let M be a 3×3 matrix satisfying
M⎡⎢⎣010⎤⎥⎦=⎡⎢⎣−123⎤⎥⎦,M⎡⎢⎣1−10⎤⎥⎦=⎡⎢⎣11−1⎤⎥⎦, and M⎡⎢⎣111⎤⎥⎦=⎡⎢⎣0012⎤⎥⎦.
Then the sum of the diagonal entries of M is
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Solution
The correct option is D 9
Clearly, M is a 3x3 matrix.
Let M=⎡⎢⎣abcdefghi⎤⎥⎦
Since, ⎡⎢⎣abcdefghi⎤⎥⎦⎡⎢⎣010⎤⎥⎦=⎡⎢⎣−123⎤⎥⎦
So, ⎡⎢⎣beh⎤⎥⎦=⎡⎢⎣−123⎤⎥⎦
⇒b=−1,e=2,h=3
Again, ⎡⎢⎣abcdefghi⎤⎥⎦⎡⎢⎣1−10⎤⎥⎦=⎡⎢⎣11−1⎤⎥⎦
So, ⎡⎢⎣a−bd−eg−h⎤⎥⎦=⎡⎢⎣11−1⎤⎥⎦
⇒a−b=1,d−e=1,g−h=−1
On solving these, we get a=0,d=3,g=2
Again, ⎡⎢⎣abcdefghi⎤⎥⎦⎡⎢⎣111⎤⎥⎦=⎡⎢⎣0012⎤⎥⎦
So, ⎡⎢⎣a+b+cd+e+fg+h+i⎤⎥⎦=⎡⎢⎣0012⎤⎥⎦
⇒a+b+c=0,d+e+f=0,g+h+i=12
⇒c=1,f=−5,i=7
So, a+e+i=0+2+7=9
Clearly, M is a 3x3 matrix.
Let M=⎡⎢⎣abcdefghi⎤⎥⎦
Since, ⎡⎢⎣abcdefghi⎤⎥⎦⎡⎢⎣010⎤⎥⎦=⎡⎢⎣−123⎤⎥⎦
So, ⎡⎢⎣beh⎤⎥⎦=⎡⎢⎣−123⎤⎥⎦
⇒b=−1,e=2,h=3
Again, ⎡⎢⎣abcdefghi⎤⎥⎦⎡⎢⎣1−10⎤⎥⎦=⎡⎢⎣11−1⎤⎥⎦
So, ⎡⎢⎣a−bd−eg−h⎤⎥⎦=⎡⎢⎣11−1⎤⎥⎦
⇒a−b=1,d−e=1,g−h=−1
On solving these, we get a=0,d=3,g=2
Again, ⎡⎢⎣abcdefghi⎤⎥⎦⎡⎢⎣111⎤⎥⎦=⎡⎢⎣0012⎤⎥⎦
So, ⎡⎢⎣a+b+cd+e+fg+h+i⎤⎥⎦=⎡⎢⎣0012⎤⎥⎦
⇒a+b+c=0,d+e+f=0,g+h+i=12
⇒c=1,f=−5,i=7
So, a+e+i=0+2+7=9
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