Question

Assertion :The determinant of a matrix $A=[a_{ij}{]}_{5\times 5}$ where $a_{ij}+a_{ij}=0$ for all i and j is zero Reason: The determinant of a skew-symmetric matrix of odd order is zero.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect but Reason is correct

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

$A=\left(\begin{array}{ccccc}{a}_{11}& a_{12}& a_{13}& a_{14}& a_{15}\\ -a_{12}& a_{22}& a_{23}& a_{24}& a_{25}\\ -a_{13}& -a_{23}& a_{33}& a_{34}& a_{35}\\ -a_{14}& -a_{24}& -a_{34}& a_{44}& a_{45}\\ -a_{15}& -a_{25}& -a_{35}& -a_{45}& a_{55}\end{array}\right)$

$A^T=\left(\begin{array}{ccccc}{a}_{11}& a_{12}& a_{13}& a_{14}& a_{15}\\ -a_{12}& a_{22}& a_{23}& a_{24}& a_{25}\\ -a_{13}& -a_{23}& a_{33}& a_{34}& a_{35}\\ -a_{14}& -a_{24}& -a_{34}& a_{44}& a_{45}\\ -a_{15}& -a_{25}& -a_{35}& -a_{45}& a_{55}\end{array}\right)=-A$

The determinant of skew -symmetric matrix of odd order is $0$

Hence, both assertion and reason are correct and reason is the correct explanation for assertion.