Question

The value of an odd order skew-symmetric determinant is
A.Perfect square
B.Negative
C.\[\pm 1\]
D.0

Answer 1

Hint: Assume an odd skew-symmetric matrix A. We know that the determinant value of skew-symmetric matrix A and the determinant value of transpose of skew-symmetric matrix A are equal to each other. That is, \[\det (A)=det({{A}^{T}})\] . Also, the transpose matrix A is equal to the negative of the matrix A. If two matrices are equal then their determinant values are also equal. So, \[\det ({{A}^{T}})=det(-A)\] . We know the formula that, \[det(-A)={{(-1)}^{n}}\det (A)\] where n is the order of the matrix A. Using this formula for \[\det (-A)\] and take n as odd. If n is odd then \[{{(-1)}^{n}}\] is equal to -1. Now, solve it further.

Complete step-by-step answer:
According to the question, it is given that we have an odd order skew-symmetric determinant.
An odd order means an odd number of rows and columns. It means we have an odd number of rows and columns in an odd skew-symmetric matrix.
Let us assume an odd skew symmetric square matrix having order n, where n is odd.
We know the property that,
\[\det (A)=det({{A}^{T}})\] ………………………..(1)
where, \[{{\text{A}}^{\text{T}}}\] is the transpose of matrix A.
We also know that for a skew-symmetric matrix, \[{{\operatorname{A}}^{T}}=-A\] .
If two matrices are equal then its determinant values are also equal, so \[\det ({{A}^{T}})=det(-A)\] …………..(2)
We know the formula that, \[det(-A)={{(-1)}^{n}}\det (A)\] where n is the order of the matrix A.
From equation (2) and the above equation, we get
\[\det ({{A}^{T}})={{(-1)}^{n}}\det (A)\] …………………(3)
Here, n is the order of the matrix A and n is odd.
\[{{(-1)}^{n}}=-1\] ………………….(4)
From equation (3) and equation (4), we have
\[\det ({{A}^{T}})={{(-1)}^{n}}\det (A)\]
\[\Rightarrow \det ({{A}^{T}})=-\det (A)\] ……………..(5)
From equation (1) and equation (5), we get
\[\det (A)=det({{A}^{T}})\]
\[\begin{align}
  & \Rightarrow \det (A)=-\det (A) \\
 & \Rightarrow 2det(A)=0 \\
 & \Rightarrow det(A)=0 \\
\end{align}\]
So, the determinant value of A is 0.
Hence, the option (D) is the correct one.

Note: In this question, we have to be careful about the step by step process to solve this question and also one might get confused because the value of n for the formula \[det(-A)={{(-1)}^{n}}\det (A)\] is not mentioned in the question. But, we don’t need the value of n here because for every odd value of n \[{{(-1)}^{n}}\] is equal to \[-1\] .