IP3U8 HOT Q1

For any \(2\times2\) matrix \(\mathbf{A} = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\), where \(a\), \(b\), \(c\) and \(d\) are real numbers, the determinant of \(\mathbf{A}\) is denoted by

\(|\mathbf{A}| = ad - bc .\)

If \(|\mathbf{A}| \neq 0\), the inverse matrix of \(\mathbf{A}\) denoted by

\[\mathbf{A}^{-1} = \frac{1}{|\mathbf{A}|} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix},\]

such that \(\mathbf{A}\mathbf{A}^{-1} = \mathbf{A}^{-1}\mathbf{A} = \mathbf{I}\), where \(\mathbf{I}\) is the identity matrix.

1. Find the inverse of: \(\mathbf{A} = \begin{pmatrix} 2 & 1 \\ 5 & -3 \end{pmatrix}\).
2. Hence, solve the system of equations
   
   \( \;\,\, 2x + y = 5\)
   \(5x - 3y = 7\)