GATE Chemical 2020 | GA Question: 9

Question

For a matrix $M=[m_{ij}]; \: i,j=1,2,3,4$, the diagonal elements are all zero and $m_{ij}=-m_{ji}$. The minimum number of elements required to fully specify the matrix is ________

• A. $0$
• B. $6$
• C. $12$
• D. $16$

Answer 1

For a matrix $M=[m_{ij}];i,j=1,2,3,4$, the diagonal elements are all zero and $m_{ij}=−m_{ji}$ (This is the property of skew-symmetric matrix).

Now, $M= \begin{bmatrix} 0 & -m_{21} & -m_{31} & -m_{41} \\  m_{21} & 0 & -m_{32} & -m_{42} \\ m_{31} & m_{32} & 0 & -m_{43} \\  m_{41}& m_{42} & m_{43} & 0    \end{bmatrix}$

Out of $16$ elements, $4$ are $0s$ and $6$ are the negative of the other $6.$ So, we need at least $6$ elements to fully specify the matrix.

So, the correct answer is $(B).$

Comments

 Lakshman Patel RJIT what is the meaning of fully specifying the matrix?

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@Lakshman Patel RJIT shouldn't the option A also be a correct option because if we consider the non-diagonal elements to be also 0 then also it will satisfy the second condition.

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@abir_banerjee

how will you take negative of zero in non diagonal?

Answer 2

Option B.
6