Problem 1
Given \(A=\left[\begin{array}{rr}7 & -1 \\ 6 & 9\end{array}\right], B=\left[\begin{array}{rr}0 & 4 \\ 3 & -2\end{array}\right],\) and \(C=\left[\begin{array}{ll}8 & 3 \\ 6 & 1\end{array}\right],\) find (a) \(A+B\) (b) \(C-A\) \((c) 3 A\) \((d) 4 B+2 C\)
Problem 1
Consider the situation of a mass layoff (i.e., a factory shuts down) where 1,200 people become unemptoyed and now begin a job search. In this case there are two states: employed (E) and unemployed (U) with an initial vector $$x_{0}^{\prime}=\left[\begin{array}{ll} E & U \end{array}\right]=\left[\begin{array}{ll} 0 & 1,200 \end{array}\right]$$ Suppose that in any given period an unemployed person will find a job with probability .7 and will therefore remain unemployed with a probability of .3. Additionally, persons who find themselves employed in any given period may lose their job with a probability of .1 (and will have a .9 probability of remaining employed). (a) Set up the Markov transition matrix for this problem. (b) What will be the number of unemployed people after (i) 2 periods; (ii) 3 periods; (iii) 5 periods; (iv) 10 periods? (c) What is the steady-state level of unemployment?
Problem 1
Given \(A=\left[\begin{array}{ll}3 & 6 \\ 2 & 4\end{array}\right], B=\left[\begin{array}{rr}-1 & 7 \\ 8 & 4\end{array}\right],\) and \(C=\left[\begin{array}{ll}3 & 4 \\ 1 & 9\end{array}\right],\) verify that \((a)(A+B)+C=A+(B+C)\) (b) \((A+B)-C=A+(B-C)\)
Problem 2
\\[ \text { Given } A=\left[\begin{array}{ll} 2 & 8 \\ 3 & 0 \\ 5 & 1 \end{array}\right], B=\left[\begin{array}{ll} 2 & 0 \\ 3 & 8 \end{array}\right], \text { and } C=\left[\begin{array}{ll} 7 & 2 \\ 6 & 3 \end{array}\right]: \\] (a) Is \(A B\) defined? Calculate \(A B\). Can you calculate \(8 A\) ? Why? (b) Is \(B C\) defined? Calculate \(B C\). Is CB defined? If so, calculate \(C B\). Is it true that \(B C=C B ?\)
Problem 2
$$\text { Siven } w=\left[\begin{array}{r} 3 \\ 2 \\ 16 \end{array}\right], x=\left[\begin{array}{l} x_{1} \\ x_{2} \end{array}\right], y=\left[\begin{array}{l} y_{1} \\ y_{2} \end{array}\right], \text { and } z=\left[\begin{array}{l} z_{1} \\ z_{2} \end{array}\right]$$ (a) Which of the following are defined: \(w^{\prime} x, x^{\prime} y^{\prime}, x y^{\prime}, y^{\prime} y, z z^{\prime}, y w^{\prime}, x \cdot y ?\) (b) Find all the products that are defined.
Problem 2
The subtraction of a matrix \(B\) may be considered as the addition of the matrix (-1)\(B\). Does the commutative law of addition permit us to state that \(A-B=B-A ?\) If not, how would you correct the statement?
Problem 3
Having sold \(n\) items of merchandise at quantities \(Q_{1}, \ldots, Q_{n}\) and prices \(P_{1}, \ldots, P_{n}\) how would you express the total revenue in \((a) \sum\) notation and (b) vector notation?
Problem 3
Ceneralize the result (4.11) to the case of a product of three matrices by proving that, for any conformable matrices \(A, B,\) and \(C,\) the equation \((A B C)^{\prime}=C^{\prime} B^{\prime} A^{\prime}\) holds.
Problem 4
Given the following four matrices, test whether any one of them is the inverse of another: \\[ D=\left[\begin{array}{rr} 1 & 12 \\ 0 & 3 \end{array}\right] \quad E=\left[\begin{array}{rr} 1 & 1 \\ 6 & 8 \end{array}\right] \quad F=\left[\begin{array}{rr} 1 & -4 \\ 0 & \frac{1}{3} \end{array}\right] \quad C=\left[\begin{array}{rr} 4 & -\frac{1}{2} \\ -3 & \frac{1}{2} \end{array}\right] \\]
Problem 4
Prove that for any two scalars \(g\) and \(k\) (a) \(k(A+B)=k A+k B\) \((b)(g+k) A=g A+k A\) (Note: To prove a result, you cannot use specific examples.)
