The matrix \( A \) is such that \( A^{2}=4 A-3 I \) where \( I \) is the corresponding identity matrix. Find integers \( p \) and \( q \) such that \[ A^{4}=p A+q I . \]

To solve for \( A^{4} \) in terms of \( A \) and \( I \), we start from the given equation: \[ A^{2} = 4A - 3I. \] We want to find \( A^{4} \). We can express \( A^{4} \) as \( (A^{2})^{2} \): \[ A^{4} = (A^{2})(A^{2}) = (4A - 3I)(4A - 3I). \] Now we will expand this product: \[ A^{4} = (4A)(4A) + (4A)(-3I) + (-3I)(4A) + (-3I)(-3I). \] Calculating each term, we find: - \( (4A)(4A) = 16A^{2} \) - \( (4A)(-3I) = -12A \) - \( (-3I)(4A) = -12A \) - \( (-3I)(-3I) = 9I \) Combining these results gives us: \[ A^{4} = 16A^{2} - 12A - 12A + 9I = 16A^{2} - 24A + 9I. \] Next, we substitute \( A^{2} \) with \( 4A - 3I \) from the original equation: \[ 16A^{2} = 16(4A - 3I) = 64A - 48I. \] Substituting this back into our expression for \( A^{4} \): \[ A^{4} = (64A - 48I) - 24A + 9I. \] Combining the terms yields: \[ A^{4} = (64A - 24A) + (-48I + 9I) = 40A - 39I. \] Thus, we have \( A^{4} = 40A - 39I \). Identifying \( p \) and \( q \) from this expression, we find: \[ p = 40, \quad q = -39. \] So the integers \( p \) and \( q \) are \( \boxed{40} \) and \( \boxed{-39} \).