Question

Let AX = B be a system of three linear equations in three variables. Then the system has
(A) a unique solution if |A| = 0
(B) a unique solution if |A| $\neq$ 0
(C) no solutions if |A| = 0 and (adj A) B $\neq$ 0
(D) infinitely many solutions if |A| = 0 and (adj A)B = 0
Choose the correct answer from the options given below:

• A. (A), (C) and (D) only
• B. (B), (C) and (D) only
• C. (B) only
• D. (B) and (C) only

Hint:

A summary of conditions for solving AX=B:

If |A| $\neq$ 0 $\rightarrow$ Unique solution (consistent).
If |A| = 0:

Calculate (adj A)B.
If (adj A)B $\neq$ 0 $\rightarrow$ No solution (inconsistent).
If (adj A)B = 0 $\rightarrow$ Infinitely many solutions (consistent).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:

This question tests the conditions for the consistency and nature of solutions for a system of linear equations \( AX = B \), where \( A \) is a square matrix. The determinant of \( A \), \( |A| \), plays a crucial role.

Step 3: Detailed Explanation:

Let's analyze the conditions for the system of equations \( AX = B \).

Case 1: \( |A| \neq 0 \) (A is non-singular)

If the determinant of the coefficient matrix is non-zero, the matrix \( A \) is invertible. The system has a unique solution given by \( X = A^{-1}B \).

- Statement (B) says the system has a unique solution if \( |A| \neq 0 \). This is true.

- Statement (A) says the system has a unique solution if \( |A| = 0 \). This is false.

Case 2: \( |A| = 0 \) (A is singular)

If the determinant is zero, the system may have no solution or infinitely many solutions. To determine which, we calculate \( (\text{adj} A)B \). The solution is given by \( X = A^{-1}B = \frac{(\text{adj} A)B}{|A|} \).

If \( (\text{adj} A)B \neq 0 \) (the zero vector), then the system is inconsistent and has no solution.

If \( (\text{adj} A)B = 0 \) (the zero vector), then the system is consistent and has infinitely many solutions.

- Statement (C) says the system has no solution if \( |A| = 0 \) and \( (\text{adj} A)B \neq 0 \). This is true.

- Statement (D) says the system has infinitely many solutions if \( |A| = 0 \) and \( (\text{adj} A)B = 0 \). This is true.

Step 4: Final Answer:

The correct statements are (B), (C), and (D). Therefore, the correct option is (2).