The system of equations x + y + z = 2 2x + 3y + z = 5 3x + 4y + 2z =…

The correct answer is Option (4) → no solution

Write the system as AX = B with

    \( \displaystyle A = \begin{bmatrix}1 & 1 & 1\\ 2 & 3 & 1\\ 3 & 4 & 2\end{bmatrix},\quad B = \begin{bmatrix}2\\ 5\\ 9\end{bmatrix} \)

Expanding along the first row:

    \( |A| = 1(6 - 4) - 1(4 - 3) + 1(8 - 9) \)

    \( = 2 - 1 - 1 = 0 \)

So there is no unique solution, and we look at \((\mathrm{adj}\,A)B\). The cofactors are

    \( C_{11} = 2,\ C_{12} = -1,\ C_{13} = -1 \)

    \( C_{21} = 2,\ C_{22} = -1,\ C_{23} = -1 \)

    \( C_{31} = -2,\ C_{32} = 1,\ C_{33} = 1 \)

so

    \( \displaystyle (\mathrm{adj}\,A)B = \begin{bmatrix}2 & 2 & -2\\ -1 & -1 & 1\\ -1 & -1 & 1\end{bmatrix}\begin{bmatrix}2\\ 5\\ 9\end{bmatrix} \)

    \( \displaystyle = \begin{bmatrix}4 + 10 - 18\\ -2 - 5 + 9\\ -2 - 5 + 9\end{bmatrix} = \begin{bmatrix}-4\\ 2\\ 2\end{bmatrix} \)

Since \(|A| = 0\) and \((\mathrm{adj}\,A)B \ne O\), the system is inconsistent: it has no solution.

Check: adding the first two equations gives \(3x + 4y + 2z = 7\), while the third equation says the same expression equals 9.