\begin{tabular}{l} \#3.) Diagonalize the matrix A , by finding matrix D and matrix P , where \\ \( \qquad A=\left[\begin{array}{cc}4 & -3 \\ 2 & -1\end{array}\right] \) \\ \hline Note: We want to express A as \( A=P D P^{-1} \quad \), where \( D \) is a diagonal matrix. \end{tabular}

To diagonalize the matrix \( A = \begin{pmatrix} 4 & -3 \\ 2 & -1 \end{pmatrix} \), we start by finding the eigenvalues. The characteristic polynomial is given by \( \det(A - \lambda I) = 0 \). Calculating it, we have: \[ A - \lambda I = \begin{pmatrix} 4 - \lambda & -3 \\ 2 & -1 - \lambda \end{pmatrix} \] Calculating the determinant: \[ (4 - \lambda)(-1 - \lambda) - (-3)(2) = \lambda^2 - 3\lambda + 4 = 0 \] Upon solving this quadratic, we find the eigenvalues: \[ \lambda^2 - 3\lambda + 4 = 0 \Rightarrow \lambda = \frac{3 \pm \sqrt{(-3)^2 - 4 \cdot 1 \cdot 4}}{2 \cdot 1} = \frac{3 \pm i\sqrt{7}}{2} \] Now, let’s find the eigenvectors corresponding to the eigenvalues \( \lambda_1 = \frac{3 + i\sqrt{7}}{2} \) and \( \lambda_2 = \frac{3 - i\sqrt{7}}{2} \). For the first eigenvalue: \[ A - \lambda_1 I = \begin{pmatrix} 4 - \frac{3 + i\sqrt{7}}{2} & -3 \\ 2 & -1 - \frac{3 + i\sqrt{7}}{2} \end{pmatrix} = \begin{pmatrix} \frac{5 - i\sqrt{7}}{2} & -3 \\ 2 & -\frac{5 + i\sqrt{7}}{2} \end{pmatrix} \] We can find the eigenvectors by solving the system given by setting the determinant equal to zero. The process involves substituting back into the equations to find the appropriate values for the eigenvector. Once we find the eigenvectors \( v_1 \) and \( v_2 \) corresponding to \( \lambda_1 \) and \( \lambda_2 \) respectively, we can construct the matrix \( P \) from these eigenvectors and the diagonal matrix \( D \) will have our eigenvalues: \[ D = \begin{pmatrix} \frac{3 + i\sqrt{7}}{2} & 0 \\ 0 & \frac{3 - i\sqrt{7}}{2} \end{pmatrix}, \quad P = \begin{pmatrix} \text{eigenvector}_1 & \text{eigenvector}_2 \end{pmatrix} \] Finally, you can compute \( P^{-1} \) using the formula for the inverse of a 2x2 matrix. Thus, we express \( A \) as \( A = P D P^{-1} \) once all components are adequately calculated.