Solve the pair of differential equations
x'_{1 } (t) = x_{1 } (t) + x_{2 } (t)  
x'_{2 } (t) =  x_{1 } (t) + x_{2 } (t)  
with initial conditions x_{1}(0) = x_{2}(0) = 1000.
Solve the initial value problem
by converting it into a system of differential equations of order 1 and then solving the system.
Use Gershgorin discs to study the location of the eigenvalues of the matrix
2j  2  1  
1  2j  0  
j  1  2  
Show that if is an eigenvalue of the matrix
2  1  0  0  
1  2  1  0  
0  1  2  1  
0  0  1  2  
Consider the matrix
A =  2  2  1  
1  3  1  
2  4  3  
Compute the norms A_{1 }, A_{} and A_{Fr } of the matrix
A =  1  1  0  1  
0  1  2  1  
1  0  1  1  
2  0  1  1  
Consider the matrix A of the previous exercise. Compute A_{2} iteratively. Start from the vector _{0 } = (0, 1,  1, 1).
Compute the condition number of A with respect to 1norm and norm. Here
A =  0.501  0.343  .  
0.872  0.597  
Compute e^{A}, where


Compute the spectral radius of the matrix
A =  1  0  0  
0  1  1  
1  1  2  