State Space Equation:
The state space representation of a system is a mathematical model that describes the behavior of a dynamic system. It consists of a set of first-order differential equations, known as state equations, that describe how the system's state variables change over time.

The general form of a state space equation is:
x'(t) = Ax(t) + Bu(t)
y(t) = Cx(t) + Du(t)

Where:
- x(t) is the state vector, representing the system's internal state at time t.
- x'(t) is the derivative of x(t) with respect to time.
- u(t) is the input vector, representing the system's inputs at time t.
- y(t) is the output vector, representing the system's outputs at time t.
- A is the state matrix, describing the dynamics of the system.
- B is the input matrix, relating the inputs to the state variables.
- C is the output matrix, relating the state variables to the outputs.
- D is the feedforward matrix, relating the inputs directly to the outputs.

x1(t) = x2(t)
In the given system, the two state variables x1(t) and x2(t) are equal. This implies that the state vector x(t) can be represented as [x1(t), x1(t)].

Controllability:
Controllability refers to the ability to manipulate the system's state variables by applying suitable inputs. A system is said to be controllable if it is possible to steer the system from any initial state to any desired state in a finite time.

To determine the controllability of a system, we need to check the rank of the controllability matrix. The controllability matrix is given by:
C = [B AB A^2B ... A^(n-1)B]

If the rank of the controllability matrix is equal to the number of states, then the system is controllable.

Observability:
Observability refers to the ability to determine the system's internal states by measuring its outputs. A system is said to be observable if it is possible to reconstruct the state vector x(t) from the output vector y(t).

To determine the observability of a system, we need to check the rank of the observability matrix. The observability matrix is given by:
O = [C; CA; CA^2; ...; CA^(n-1)]

If the rank of the observability matrix is equal to the number of states, then the system is observable.

Uncontrollable System:
In the given system, x1(t) = x2(t), the state vector can be represented as [x1(t), x1(t)]. The state matrix A will be a diagonal matrix with repeated eigenvalues. The input matrix B will have one input for each state variable.

Since the state variables are equal, the controllability matrix will be:
C = [B AB A^2B ... A^(n-1)B]
[B AB A^2B ... A^(n-1)B]

The rank of the controllability matrix will be less than the number of states (n), as the rows are repeated. Therefore, the system is uncontrollable.

Conclusion:
The given system is uncontrollable because the state variables are equal, resulting

B