1. \( -3 p+q=0 \) \( 2 p-q=1 \) Solve the following simultaneous linear equations using the matrix method.
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To solve the simultaneous equations using the matrix method, we can represent the equations in matrix form as \( AX = B \), where \[ A = \begin{pmatrix} -3 & 1 \\ 2 & -1 \end{pmatrix}, X = \begin{pmatrix} p \\ q \end{pmatrix}, B = \begin{pmatrix} 0 \\ 1 \end{pmatrix} \] We can calculate the inverse of matrix \( A \) and then find \( X \) using \( X = A^{-1}B \). The determinant of \( A \) is \( (-3)(-1) - (2)(1) = 3 - 2 = 1 \), which is non-zero, so we can find the inverse: \[ A^{-1} = \frac{1}{\text{det}(A)} \begin{pmatrix} -1 & -1 \\ 2 & -3 \end{pmatrix} = \begin{pmatrix} -1 & -1 \\ 2 & -3 \end{pmatrix} \] Now, multiplying \( A^{-1} \) by \( B \): \[ X = \begin{pmatrix} -1 & -1 \\ 2 & -3 \end{pmatrix} \begin{pmatrix} 0 \\ 1 \end{pmatrix} = \begin{pmatrix} -1 \\ -3 \end{pmatrix} \] Thus, we find \( p = -1 \) and \( q = -3 \).
