The sum of three numbers x, y and z is 9. When the second number is subtracted from the sum of the first and the third, the result is 3. When the third number is subtracted from the sum of twice the first and the second, the result is 3. Solving these equations by the matrix method, the product xyz is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 24
The conditions give the system
\(x + y + z = 9\)
\(x - y + z = 3\)
\(2x + y - z = 3\)
In matrix form AX = B:
\(\displaystyle A = \begin{bmatrix}1 & 1 & 1\\ 1 & -1 & 1\\ 2 & 1 & -1\end{bmatrix},\quad B = \begin{bmatrix}9\\ 3\\ 3\end{bmatrix}\)
\(|A| = 1(1 - 1) - 1(-1 - 2) + 1(1 + 2) = 6\)
The cofactors are \(C_{11} = 0, C_{12} = 3, C_{13} = 3, C_{21} = 2, C_{22} = -3, C_{23} = 1, C_{31} = 2, C_{32} = 0, C_{33} = -2\). The adjoint is their transpose:
\(\displaystyle A^{-1} = \frac{1}{6}\begin{bmatrix}0 & 2 & 2\\ 3 & -3 & 0\\ 3 & 1 & -2\end{bmatrix}\)
\(\displaystyle X = A^{-1}B = \frac{1}{6}\begin{bmatrix}0 + 6 + 6\\ 27 - 9 + 0\\ 27 + 3 - 6\end{bmatrix} = \frac{1}{6}\begin{bmatrix}12\\ 18\\ 24\end{bmatrix}\)
\(x = 2,\ y = 3,\ z = 4\)
\(xyz = 2 \times 3 \times 4 = 24\)
Check: \(2 + 3 + 4 = 9\), \(2 + 4 - 3 = 3\), \(4 + 3 - 4 = 3\).