I would like to solve a $ 2 times $ 2 system of form

$$ frac {d} {d theta} T = TA, quad T (0) = Id $$

or $ theta $ is real and $ A $ is of the form

$$ A = begin {pmatrix} 0 & frac {e ^ {- i theta}} { lambda} \ frac {1} {36} e ^ {- i theta} left (9 lambda + 2 ( lambda-1) ^ 2 (6 cos { theta} + cos {2 theta} + 6) right) & 0 end {pmatrix}, $$

with $ lambda $ a free parameter in the **unitary circle**.

In particular, I'm interested in getting digital solutions to $ theta = 2 pi $ depending on the additional parameter $ lambda $. I'm pretty new using *Mathematica*and that's what I've tried so far:

```
T[θ_] = {{T11[θ], T12[θ]}, {T21[θ], T22[θ]}};
A[θ_] = {{0, E ^ (- I θ) / λ}, {1/36 E ^ (- I θ) (9 λ + 2 (-1 + λ) ^ 2 (6 + 6 Cos[θ] + Cos[2 θ])), 0}};
sys = {T & # 39;[θ] == T[θ].A[θ]};
```

The previous code defines the system I want to solve and I am now trying to solve it numerically. I first tried

```
NSol = NDSolve[{sys, T11[0] == 1, T12[0] == 0, T21[0] == 0, T22[0] == 1}, {T11[θ], T12[θ], T21[θ], T22[θ]}, {θ}, {θ, 0, 2 Pi}];
```

which gives me the exit

```
NDSolve :: dupv: "Double variable θ found in NDSolve[<[<[<[<<1>>]. "
```

I have also tried

```
Nsol2 = ParametricNDSolve[{sys, T11[0] == 1, T12[0] == 0, T21[0] == 0, T22[0] == 1}, {T11, T12, T21, T22}, {θ, 0, 2 Pi}, {λ}];
```

which gives me exit $ T_ {11}, dots, T_ {22} $ as *ParametricFunction*s dependent on each other and on $ lambda $.

I do not know if this is the right approach and, if so, how to extract a numeric expression based on $ lambda $ of the last output – all that I saw in the documentation are examples plotted for specific values of the parameter. Any help is very appreciated.