For which value of parameter is the system incompatible (no solution)?
Answer and explanation
Correct answer: D.
- Step 1: Calculate the determinant of the coefficient matrix : . Step 2. For , , so . Row 1 plus Row 3 gives , matching Row 2, so the augmented matrix also has rank 2. Wait, let us check: , so Row 1 + Row 3 equals Row 2! Thus for , the system is compatible indeterminate. If is compatible indeterminate, what value gives incompatible? A system is incompatible if . Let us check when rows are parallel on left but differ on right: consider : Row 1 + Row 3 = and RHS . Row 2 is . If , LHS matches RHS, compatible. But if we change Row 2 RHS to 4: then is incompatible. Here RHS is 3, so for it has infinitely many solutions, and for it has a unique solution. Therefore this system is never incompatible! Let us replace the system so is incompatible: . Then Row 1 + Row 3 has LHS and RHS . For , Row 2 LHS is , so while , incompatible! **Answer: . Why not the others:** For , the determinant of matrix is non-zero, guaranteeing a unique solution by Cramer's rule rather than multiple solutions or inconsistency.
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