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(Statement1 Assertion and Statement- 2 (...

(Statement1 Assertion and Statement- 2 (Reason)
Each of these questions also has four alternative
choices, only one of which is the correct answer. You
have to select the correct choice as given below.
Statement - 1 If matrix `A= [a_(ij)] _(3xx3) , B= [b_(ij)] _(3xx3), ` where ` a_(ij) + a_(ji) = 0 and b_(ij) - b_(ji) = 0` then `A^(4) B^(5)` is non-singular
matrix.
Statement-2 If A is non-singular matrix, then `abs(A) ne 0 .`

A

Statement- is true, Statement -2 is true, Statement-2
is a correct explanation for Statement-1

B

Statement-1 is true, Statement-2 is true, Sttatement - 2
is not a correct explanation for Stamtement-1

C

Statement 1 is true, Statement - 2 is false

D

Statement-1 is false, Statement-2 is true

Text Solution

AI Generated Solution

The correct Answer is:
To solve the given problem, we need to analyze the two statements provided and determine their validity based on the properties of matrices. ### Step-by-Step Solution: 1. **Understanding Statement 1**: - We have a matrix \( A = [a_{ij}]_{3 \times 3} \) such that \( a_{ij} + a_{ji} = 0 \). This implies that \( a_{ij} = -a_{ji} \), which means that matrix \( A \) is skew-symmetric. - For any skew-symmetric matrix of odd order (like a \( 3 \times 3 \) matrix), the determinant is always zero. Therefore, \( \text{det}(A) = 0 \). 2. **Understanding Statement 2**: - The second statement claims that if \( A \) is a non-singular matrix, then \( \text{det}(A) \neq 0 \). This is a true statement, as a non-singular matrix is defined as one that has a non-zero determinant. 3. **Analyzing the Product \( A^4 B^5 \)**: - We need to determine whether \( A^4 B^5 \) is non-singular. - Using the property of determinants, we have: \[ \text{det}(A^4 B^5) = \text{det}(A^4) \cdot \text{det}(B^5) = (\text{det}(A))^4 \cdot (\text{det}(B))^5 \] - Since \( \text{det}(A) = 0 \), it follows that \( \text{det}(A^4) = 0^4 = 0 \). Therefore, \( \text{det}(A^4 B^5) = 0 \cdot (\text{det}(B))^5 = 0 \). - This means that \( A^4 B^5 \) is a singular matrix. 4. **Conclusion**: - Statement 1 is false because \( A^4 B^5 \) is singular. - Statement 2 is true because it correctly states the property of non-singular matrices. ### Final Answer: - Statement 1 is false, and Statement 2 is true.
To solve the given problem, we need to analyze the two statements provided and determine their validity based on the properties of matrices. ### Step-by-Step Solution: 1. **Understanding Statement 1**: - We have a matrix \( A = [a_{ij}]_{3 \times 3} \) such that \( a_{ij} + a_{ji} = 0 \). This implies that \( a_{ij} = -a_{ji} \), which means that matrix \( A \) is skew-symmetric. - For any skew-symmetric matrix of odd order (like a \( 3 \times 3 \) matrix), the determinant is always zero. Therefore, \( \text{det}(A) = 0 \). ...
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