- If a 4x4 matrix has a
$det(A)=1/2$ , find$det(2A)$ ,$det(-A)$ ,$det(A^2)$ , and$det(A−1)$ . (10 pts.) - Consider matrix: $A=\begin{bmatrix}
1 & 2 & 0 \
2 & 6 & 4 \
0 & 4 & 11
\end{bmatrix}$. Find the symmetric factorization
$𝐴=LDL^T$ , Find$𝐴^{-1}$ (10 pts.) - Find a parabola that best fits to the following points: (10 pts.) $$ (-1, 2), (1, -3), (0, 0), (2, - 5). $$
- Find extrema and saddle points for the following function: (10 pts.) $$ f(x,y)=3x^3+y^2+4xy-x+2 $$ Formulate Sylvester’s criterion (2 pts.).
- Prove that for any square matrix
$A(n\times n)$ with eigenvalues${\lambda_1, \lambda_2, ..., \lambda_n}$ the multiplication:$(A-\lambda_1I)(A-\lambda_2I)...(A-\lambda_nI)$ or produces the zero matrix? (12 pts.) - Find
$A^{10}$ for the matrix $A=\begin{bmatrix} 4&3\ 1&2 \end{bmatrix}$ (10 pts.) - Find eigenvector of the circulant matrix
$C$ for the eigenvalue$\lambda=c_1+c_2+c_3+c_4$ (12 pts.) $$ C=\begin{bmatrix} c_1 & c_2 & c_3 & c_4 \ c_4 & c_1 & c_2 & c_3 \ c_3 & c_4 & c_1 & c_2 \ c_2 & c_3 & c_4 & c_1 \end{bmatrix} $$ - Solve the differential equation, (10 pts.):
$$
\frac{d\vec{u}}{dt}=\begin{bmatrix}4&-2\1&1\end{bmatrix}\vec{u}(t), \vec{u}(0)=\begin{bmatrix}3\2\end{bmatrix}
$$
What happens to
$\vec{u}(t)$ as$t\rightarrow \infty$ ? (2 pts.) - Apply the Gram-Schmidt process to:
$$
x_1=\begin{bmatrix}0\0\2\0\end{bmatrix}, x_2=\begin{bmatrix}-1\0\1\0\end{bmatrix}, x_3=\begin{bmatrix}2\1\0\0\end{bmatrix}, x_4=\begin{bmatrix}1\0\0\-1\end{bmatrix}
$$
and write the result in form
$A=QR$ . (12 pts.) - Find the SVD and the pseudoinverse of the matrix: $A=\begin{bmatrix} 1 & 0 & 1 \ 0 & 1 & 0 \end{bmatrix}$. (10 pts.)