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**Unformatted text preview: **Show that QXs = Xs This is called the steady state because the number of people
who get each paper didn’t change for the next week. Show that Qn X → Xs as n → ∞.
(f) Now let
S= 2
3
1
3 2
3
1
3 Show that Qn → S as n → ∞.
(g) Show that SY = Xs for any matrix Y of the form
Y= y
150 − y 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.
10 More speciﬁcally, we have a Markov Chain, which is a special type of stochastic process. 8.3 Matrix Arithmetic 491 4. Let z = a + bi and w = c + di be arbitrary complex numbers. Associate z and w with the
matrices
ab
cd
Z=
and W =
−b a
−d c
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.
5. 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, 1
2
3
4...

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