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Let \(A = \left [ \begin{array}{rrr} a & b & c \\ d & e & f \end{array} \right ] \;\;\; E_1 = \left [ \begin{array}{rr} 0 & 1 \\ 1 & 0 \end{array} \right ] \;\;\; E_2 = \left [ \begin{array}{rr} 5 & 0 \\ 0 & 1 \end{array} \right ] \;\;\; E_3 = \left [ \begin{array}{rr} 1 & -2 \\ 0 & 1 \end{array} \right ]\)
Compute \(E_1A\), \(\; E_2A\) and \(E_3A\). What effect did each of the \(E_{i}\) matrices have on the rows of \(A\)? Create \(E_4\) so that its effect on \(A\) is to multiply the
bottom row by \(-6\). How would you extend this idea to matrices with more than two rows?
\(E_1A = \left [ \begin{array}{rrr} d & e & f \\ a & b & c\end{array} \right ]\;\;\) \(E_1\) interchanged \(R1\) and \(R2\) of \(A\). \(E_2A = \left [ \begin{array}{rrr} 5a & 5b & 5c \\ d & e & f \end{array} \right ]\;\;\) \(E_2\) multiplied \(R1\) of \(A\) by 5. \(E_3A = \left [ \begin{array}{rrr} a - 2d & b - 2e & c - 2f \\ d & e & f \end{array} \right ]\;\;\) \(E_3\) replaced \(R1\) in \(A\) with \(R1 - 2R2\). \(E_4 = \left [ \begin{array}{rr} 1 & 0 \\ 0 & -6 \end{array} \right ]\;\;\)
In Exercises MCfirst - MClast, consider the following scenario. In the small village of Pedimaxus in the country of Sasquatchia, all 150
residents get one of the two local newspapers. Market research has shown that in any given week, 90% of those who
subscribe to the Pedimaxus Tribune want to keep getting it, but 10% want to switch to the Sasquatchia Picayune. Of those
who receive the Picayune, 80% want to continue with it and 20% want switch to the Tribune. We can express this situation
using matrices. Specifically, let \(X\) be the ‘state matrix’ given by
\[X = \left [ \begin{array}{r} T \\ P \end{array} \right ]\]
where \(T\) is the number of people who get the Tribune and \(P\) is the
number of people who get the Picayune in a given week. Let \(Q\) be the ‘transition matrix’ given by
Since 10% of that 150 want to switch to the Picayune, we should have that for Week 1, 135 people get the Tribune and
15 people get the Picayune. Show that \(QX\) in this situation is indeed
Assuming that the percentages stay the same, we can get to the subscription numbers for Week 2 by computing \(Q^{2}X\). How
many people get each paper in Week 2?
Explain why the transition matrix does what we want it to do.
If the conditions do not change from week to week, then \(Q\) remains the same and we have what’s known as a Stochastic
Process (More specifically, we have a Markov Chain, which is a special type of stochastic process.) because
Week \(n\)’s numbers are found by computing \(Q^{n}X\). Choose a few values of \(n\) and, with the help of your classmates
and calculator, find out how many people get each paper for that week. You should start to see a pattern as \(n \rightarrow \infty \).
If you didn’t see the pattern, we’ll help you out. Let
Show that \(QX_{s} = X_{s}\) This is called the steady state because the number of
people who get each paper didn’t change for the next week. Show that \(Q^{n}X \rightarrow X_{s}\) as \(n \rightarrow \infty \).
Show that \(Q^{n} \rightarrow S\) as \(n \rightarrow \infty \).
Show that \(SY = X_{s}\) for any matrix \(Y\) of the form
\[Y = \left [ \begin{array}{r} y \\ 150 - y \end{array} \right ]\]
This means that no matter how the distribution starts in Pedimaxus, if \(Q\) is
applied often enough, we always end up with 100 people getting the Tribune and 50 people getting the Picayune.
Let \(z = a + bi\) and \(w = c + di\) be arbitrary complex numbers. Associate \(z\) and \(w\) with the matrices
\[Z = \left [ \begin{array}{rr} a & b \\ -b & a \end{array} \right ] \;\; \mbox {and} \;\; W = \left [ \begin{array}{rr} c & d \\ -d & c \end{array} \right ]\]
Show that complex number addition, subtraction
and multiplication are mirrored by the associated matrix arithmetic. That is, show that \(Z + W\), \(Z - W\) and \(ZW\) produce matrices which can be
associated with the complex numbers \(z + w\), \(z - w\) and \(zw\), respectively.
Compare \((A + B)^{2}\) to \(A^{2} + 2AB + B^{2}\). Discuss with your classmates what constraints must be placed on two arbitrary matrices \(A\) and \(B\) so that
both \((A + B)^{2}\) and \(A^{2} + 2AB + B^{2}\) exist. When will \((A + B)^{2} = A^{2} + 2AB + B^{2}\)? In general, what is the correct formula for \((A + B)^{2}\)?
In Exercises triangexfirst - triangexlast, consider the following definitions. A square matrix is said to be an upper triangular matrix if all of its
entries below the main diagonal are zero and it is said to be a lower triangular matrix if all of its entries above the
main diagonal are zero. For example,
