1. Figure P1 shows three chemical reactors linked by pipes. As indicated, the rate of transfer of chemicals through each pipe is equal to a flow rate (Q, with units of cubic meters per second) multiplied by the concentration of the reactor from which the pipe originates (c, with units of milligrams per cubic meter). If the system is at a steady state, the transfer into each reactor will balance the transfer out. Develop mass balance equations for the reactors, and use Gaussian elimination to solve the three simultaneous linear algebraic equations for the unknown concentrations c₁, c₂, and c₃.

2. Let

(a) Find the condition that α, β, and γ must satisfy so that det P = det Q.

(b) Find Q^-1 in terms of α, β, and γ. State the condition for which Q^-1 exists.

3. Suppose

where a and b are arbitrary real numbers. Find the values of a and b, if any, such that

(a) b is not in the column space of A;

(b) the nullity of A is equal to 1;

(c) the nullity of A is equal to 2.

4. Let the linear transformation f : R³ |→ R³ be defined by

f(1, 0, 0) = (1, 0, 0),          f(1, 1, 0) = (0, 1, 0),         and        f(1, 1, 1) = (0, 0, 1).

(a) Find the matrix representation of f with respect to the standard basis in R³.

(b) Find f(1, 2, 3).

(c) Find all vectors v = [x y z]^T in R³ such that f(x, y, z) = (0, 0, 0).

5. Consider the following 4 × 4 matrix

In MATLAB you would obtain this matrix as eye(4) - ones(4).

(a) Find all the eigenvalues of A, including multiplicities.

(b) For each of the eigenvalues you found in part (a), find a corresponding set of eigenvectors.

(c) Is A diagonalizable? Why or why not? Provide a complete explanation.