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: Conceptual Foundation:

This inquiry assesses the criteria for consistency and solution types in a linear system \( AX = B \), where \( A \) is a square matrix. The determinant of \( A \), denoted \( |A| \), is pivotal.

Step 3: In-depth Analysis:

We shall examine the conditions governing the system of equations \( AX = B \).

Scenario 1: \( |A| eq 0 \) (Non-singular A)

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

- Statement (B) asserts that the system has a unique solution when \( |A| eq 0 \). This is correct.

- Statement (A) posits that the system has a unique solution when \( |A| = 0 \). This is incorrect.

Scenario 2: \( |A| = 0 \) (Singular A)

A zero determinant implies that the system may have either no solution or infinitely many solutions. To differentiate, we compute \( (\text{adj} A)B \). The solution is derived from \( X = A^{-1}B = \frac{(\text{adj} A)B}{|A|} \).

If \( (\text{adj} A)B eq 0 \) (i.e., not the zero vector), the system is inconsistent and has no solution.

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

- Statement (C) indicates no solution when \( |A| = 0 \) and \( (\text{adj} A)B eq 0 \). This is correct.

- Statement (D) specifies infinitely many solutions when \( |A| = 0 \) and \( (\text{adj} A)B = 0 \). This is correct.

Step 4: Conclusion:

Statements (B), (C), and (D) are accurate. Consequently, the correct selection is (2).