Final answer:
The question seeks a sufficient and necessary condition for the existence of a specific 2×2 matrix with real numbers a, b, c, and d. Without full context, it's speculative but such conditions often relate to the properties like determinant for invertibility or relationships between elements for eigenvalues. The provided solution relates more to kinematics than matrices.
Step-by-step explanation:
Sufficient and Necessary Conditions for a 2×2 Matrix
To derive a sufficient and necessary condition on the given real numbers a, b, c, and d that guarantees the existence of a 2×2 matrix (where the matrix is denoted with elements x, y, z, w), we need to establish a criterion that these elements must satisfy.
The question seems to involve kinematic equations, which relate to physics more than to matrix theory. Without the full context, it's challenging to give a completely accurate answer. But typically, a necessary and sufficient condition might refer to the determinant of the matrix being non-zero for invertibility or specific relationships between elements for eigenvalues.
Solution for (b): The fragments provided reference kinematic equations, particularly the equation w² = wo² + 2a0, where w represents the final velocity, wo is the initial velocity, a is the acceleration, and o appears to be a typo which typically stands for time but is not included in the function. Here, since t (time) is absent, it suggests that the matrix may relate to an instantaneous change, where time does not factor into the condition. However, due to the lack of context, this is purely speculative.