Consider the system of equations
\(x + y + z = 6\)
\(x - y + z = 2\)
\(3x + y - z = 2\)
Let A be its coefficient matrix. Consider the following statements:
(A) \(|A| = 8\)
(B) The system has infinitely many solutions.
(C) In the solution, \(y = 2\).
(D) In the solution, \(z = 3\).
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A), (C) and (D) only
The determinant of the coefficient matrix, expanded along the first row:
\( \displaystyle |A| = \begin{vmatrix}1 & 1 & 1\\ 1 & -1 & 1\\ 3 & 1 & -1\end{vmatrix} \)
\(|A| = 1(1 - 1) - 1(-1 - 3) + 1(1 + 3)\)
\(|A| = 0 + 4 + 4 = 8\)
Since \(|A| \ne 0\), the system has a unique solution. Subtracting the second equation from the first:
\(2y = 4\)
\(y = 2\)
Then the first and third equations give
\(x + z = 4\)
\(3x - z = 0\)
Adding: \(4x = 4\), so \(x = 1\) and \(z = 3\).
So \(|A| = 8\) and the unique solution is \(x = 1,\ y = 2,\ z = 3\).
Checking the statements
(A) \(|A| = 8\) — Correct. Expansion gives \(0 + 4 + 4 = 8\).
(B) Infinitely many solutions — Incorrect. \(|A| = 8 \ne 0\), so the solution is unique.
(C) \(y = 2\) — Correct. Check in the third equation: \(3(1) + 2 - 3 = 2\).
(D) \(z = 3\) — Correct. Check in the first equation: \(1 + 2 + 3 = 6\).