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

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 \( T(-2,4,-1) \). Select one:

\[ F(x, y, z)=(x-y, z) \] . 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 \( U=\mathcal{L}((1,0,0),(0,2,0) \).

[The most general such \( R \) is \( U \sqrt{A} \), where \( U \) is some isometry.] 6- Let \( x_{1}, \ldots, x_{n} \) be \( n \) vectors. Let \( G \) be a matrix with elements \( G_{i j}=\left(x_{i}, x_{j}\right) \). Show that \( G \) is a positive matrix. [ \( G \) is called the Gram matrix.]

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

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