Question

Assertion: Determinant of a skew-symmetric matrix of order 3 is zero.
Reason: For any matrix A, \[\left| A \right|=\left| {{A}^{T}} \right|\] and \[\left| -A \right|=-\left| A \right|\]where, \[\left| A \right|\]denotes the determinant of matrix A. Then,
A. Both the Assertion and Reasons are correct and Reason is the correct explanation for Assertion
B. Both the Assertion and Reasons are correct but and Reason is not correct explanation for Assertion
C. Assertion is the correct but Reason is incorrect
D. Both Assertion and reasons are correct.

Answer 1

Hint: We address this difficulty by examining each assertion and explanation separately. We employ some requirements for assertion and reasoning, such as any matrix being skew symmetric if and only if it is the inverse of its transpose, and the generic properties of determinants is given as \[\left| A \right|=\left| {{A}^{T}} \right|\]and \[\left| -A \right|={{\left( -1 \right)}^{n}}\left| {{A}^{T}} \right|\] where n is the number of rows or columns of a square matrix. Use these two properties to reach the answer.

Complete step-by-step solution:
For Assertion,
Let A be a skew- symmetric matrix of \[n\times n\] order, where n is 3.
We know that the determinant of A is always equal to the determinant of its transpose.
\[\left| A \right|=\left| {{A}^{T}} \right|---(1)\]
However, since A is a skew-symmetric matrix were,
\[{{a}_{ij}}=-{{a}_{ji}}\] (i and j are rows and column numbers).
Therefore, in case of skew-symmetric matrix
\[\left| {{A}^{T}} \right|={{\left( -1 \right)}^{n}}\left| A \right|---(2)\]
But in the assertion, it is given that \[n=3\]
\[\Rightarrow \left| {{A}^{T}} \right|={{\left( -1 \right)}^{3}}\left| A \right|\]
As we know that \[{{(-1)}^{3}}=-1\], we get:
\[\Rightarrow \left| {{A}^{T}} \right|=\left( -1 \right)\left| A \right|\]
Substituting the value of \[\left| {{A}^{T}} \right|\] in equation (1), we have
\[\Rightarrow \left| A \right|=-\left| A \right|\]
By rearranging the term, we get:
\[\Rightarrow \left| A \right|+\left| A \right|=0\]
By simplifying this further we get:
\[\Rightarrow 2\left| A \right|=0\]
From this above step we get the value of \[\left| A \right|\]that is
\[\Rightarrow \left| A \right|=0\]
Hence, the determinant of an odd skew- symmetric matrix is always zero
For Reasoning we have the equation (2) that is \[\left| {{A}^{T}} \right|={{\left( -1 \right)}^{n}}\left| A \right|\]
Here, substitutes the value of equation (1) on equation (2) we get:
\[\Rightarrow \left| A \right|={{\left( -1 \right)}^{n}}\left| A \right|---(3)\]
Now, consider the value of \[n=3\]and substitute in the equation (3) we get:
\[\Rightarrow \left| A \right|={{\left( -1 \right)}^{3}}\left| A \right|\]
As we know that \[{{(-1)}^{3}}=-1\], we get:
\[\Rightarrow \left| A \right|=-\left| A \right|\]
Hence, we can see that \[\left| A \right|=-\left| A \right|\] This condition holds true when the value of n is odd.
Reasoning is incorrect because \[\left| A \right|=-\left| A \right|\] for odd matrices only.
Hence, Assertion is correct but Reason is incorrect.
So, the correct option is “option C”.

Note: Remember all of the matrix's properties in order to tackle these types of problems. The following are some of the characteristics of a skew symmetric matrix: A skew-symmetric matrix is a scalar multiple of a skew-symmetric matrix. A skew-symmetric matrix's members on the diagonal are all zero, hence its trace is also zero. The sum of the matrix's diagonal elements is its trace.