Question

In solving a system of linear equations \(AX = B\) by Cramer's rule, in the usual notation, if \[ \Delta_1 = \begin{vmatrix} -11 & 1 & -7\\ -4 & 1 & -2 \\ 5 & 1 & 1 \end{vmatrix} \quad \text{and} \quad \Delta_3 = \begin{vmatrix} 4 & 1 & -11 \\ 3 & 1 & -4 \\ 4 & 1 & 5 \end{vmatrix}, \quad \text{then } X = ? \]

• A. \(\begin{bmatrix} -1 \\ 1 \\ 2 \end{bmatrix}\)
• B. \(\begin{bmatrix} 2 \\ 1 \\ -1 \end{bmatrix}\)
• C. \(\begin{bmatrix} 1 \\ -1 \\ 2 \end{bmatrix}\)
• D. \(\begin{bmatrix} 1 \\ 2 \\ -1 \end{bmatrix}\)

Hint:

Cramer's rule expresses the solution of a linear system in terms of determinants; replacing columns in the coefficient matrix with the constants vector gives each component of the solution.

Answer 1

The Correct Option is C

By Cramer's rule, for a system \(AX = B\), the solution vector \(X\) is given by \[ X = \begin{bmatrix} \frac{\Delta_1}{\Delta}
\frac{\Delta_2}{\Delta}
\frac{\Delta_3}{\Delta} \end{bmatrix}, \] where \(\Delta = \det(A)\) and \(\Delta_i\) is the determinant of matrix \(A\) with the \(i^{th}\) column replaced by vector \(B\). Given the determinants \(\Delta_1\) and \(\Delta_3\), and by evaluating the determinants (or given the relation in the problem), the solution vector is found to be \[ X = \begin{bmatrix} 1 \\ -1 \\ 2 \end{bmatrix}. \]