Since ∣A∣=0, the matrix is non-singular. A non-singular matrix always results in a unique solution for the system AX=B.
Assertion A is True.
Step 2: Analyze Reason R
Reason R states that if A is a 3×3 matrix and B is a non-zero column matrix, then AX=B has a unique solution if A is non-singular.
According to Cramer's Rule and Matrix Theory, if ∣A∣=0 (meaning A is non-singular), the inverse A−1 exists. The solution is given by:
X=A−1B
This solution is unique.
Reason R is True.
Conclusion
Both statements are correct, and Reason R provides the mathematical justification for why Assertion A is true (because the determinant we calculated was non-zero).
Correct Answer:
Both A and R are true and R is the correct explanation of A.
Explanation
Step 1: Analyze Assertion A
To determine if the system has a unique solution, we calculate the determinant of the coefficient matrix A.
Since ∣A∣=0, the matrix is non-singular. A non-singular matrix always results in a unique solution for the system AX=B.
Assertion A is True.
Step 2: Analyze Reason R
Reason R states that if A is a 3×3 matrix and B is a non-zero column matrix, then AX=B has a unique solution if A is non-singular.
According to Cramer's Rule and Matrix Theory, if ∣A∣=0 (meaning A is non-singular), the inverse A−1 exists. The solution is given by:
X=A−1B
This solution is unique.
Reason R is True.
Conclusion
Both statements are correct, and Reason R provides the mathematical justification for why Assertion A is true (because the determinant we calculated was non-zero).
Correct Answer:
Both A and R are true and R is the correct explanation of A.
