To determine the correct matrices $D$ and $P$ in the diagonalization of matrix $A$, we need to identify:

1. The eigenvalues of $A$, which will make up the entries in $D$.
2. The corresponding eigenvectors of $A$, which will make up the columns of $P$.

Given: $A = \begin{pmatrix} 5 & 3 \\ -4 & -2 \end{pmatrix}$

Step 1: Finding Eigenvalues

The eigenvalues $\lambda$ are solutions to the characteristic polynomial: $\det(A - \lambda I) = 0$ Calculate $\det(A - \lambda I)$: $A - \lambda I = \begin{pmatrix} 5 - \lambda & 3 \\ -4 & -2 - \lambda \end{pmatrix}$ The determinant is: $(5 - \lambda)(-2 - \lambda) - (3)(-4) = \lambda^2 - 3\lambda - 2$ Solving gives eigenvalues