
The determinant of an odd order skew symmetric matrix is always:
A. Zero
B. One
C. Negative
D. Depends on the matrix
Answer
443.2k+ views
Hint: A matrix is skew- symmetric if and if it is the opposite of its transpose and the general properties of determinants is given as where is number of rows or columns of square matrix. Use these two properties to reach the answer.
Complete Step-by-Step solution:
Let A be a skew- symmetric matrix of order, where is odd
We know that the determinant of A is always equal to the determinant of its transpose.
However, since A is a skew-symmetric matrix where
are rows and column numbers).
Therefore, in case of skew-symmetric matrix
Substituting the value of in equation (1), we have
Hence, the determinant of an odd skew- symmetric matrix is always zero and the correct option is A.
Note: In order to solve these types of questions, remember all the properties of the matrix. Some of the properties of skew symmetric matrix are - A scalar multiple of a skew symmetric matrix is skew- symmetric matrix. The elements on the diagonal of a skew-symmetric matrix are zero, and therefore its trace equals to zero. Trace of the matrix is the sum of its diagonal elements.
Complete Step-by-Step solution:
Let A be a skew- symmetric matrix of
We know that the determinant of A is always equal to the determinant of its transpose.
However, since A is a skew-symmetric matrix where
Therefore, in case of skew-symmetric matrix
Substituting the value of
Hence, the determinant of an odd skew- symmetric matrix is always zero and the correct option is A.
Note: In order to solve these types of questions, remember all the properties of the matrix. Some of the properties of skew symmetric matrix are - A scalar multiple of a skew symmetric matrix is skew- symmetric matrix. The elements on the diagonal of a skew-symmetric matrix are zero, and therefore its trace equals to zero. Trace of the matrix is the sum of its diagonal elements.
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