4. Hallar los autovalores reales \( y \) autovectores de \( \mathbb{R}^{n}, y \) una base de au- tovectores cuando sea posible, de las matrices: a) \( \left(\begin{array}{rr}4 & 6 \\ -3 & -5\end{array}\right) \), b) \( \left(\begin{array}{cc}5 & -1 \\ -1 & 1\end{array}\right) \), c) \( \left(\begin{array}{cc}1 & -2 \\ -2 & 3\end{array}\right) \),d) \( \left(\begin{array}{rrr}2 & 2 & -1 \\ 0 & -2 & 0 \\ -1 & 0 & 0\end{array}\right) \)

Use ajdoint matrix to evaluate the inverse of matrix \( A=\left(\begin{array}{cccc}3 & -3 & 2 & 4 \\ -4 & 4 & -2 & -2 \\ 2 & -3 & -3 & -3 \\ -1 & 2 & -4 & -3\end{array}\right) \)

31. For two vectors \( \vec{A} \) and \( \vec{B} \), if \( \vec{A} \times \vec{B}=6 \hat{\imath}+2 \hat{j}+5 \hat{k}, \vec{A} \cdot \vec{B}=-13, \vec{A}+\vec{B}=2 \hat{i}+\hat{j}+2 \vec{k} \) then find: \( |\vec{A}|=3 \), the vectors \( \vec{A} \) and \( \vec{B} \) a) The two b) The area of the parallelogram and the triangle having \( \vec{A} \) and \( \vec{B} \) as their edge. c) A third vector having magnitude twice \( \vec{A} \) and direction perpendicular to both vectors.

Let \( T: R^{\wedge} 3 \rightarrow R^{\wedge} 3 \) be a linear transformation such that \( T(1,0,0)=(2,4,-1), T(0,1,0)=(1,3,-2), T(0,0,1)=(0,-2,2) \). Compute \( T(-2,4,-1) \).

\[ F(x, y, z)=(x-y, z) \] . Determinare la controimmagine di \( (2,2) \) sotto l'azione di \( F \). (i) Determinare una base e la dimensione di Im \( F \) e di Ker \( F \). (ii) Calcolare la dimensione dell'immagine del sottospazio \( U=\mathcal{L}((1,0,0),(0,2,0) \).

Find the rank of matrix A by using the row echelon form. \( A=\left[\begin{array}{lll}1 & 2 & 3 \\ 2 & 1 & 4 \\ 3 & 0 & 5\end{array}\right] \)