91Ó°ÊÓ

Use elementary matrices to find the inverse of $$ \begin{array}{l} A=\left[\begin{array}{lll} 1 & a & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]\left[\begin{array}{lll} 1 & 0 & 0 \\ b & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]\left[\begin{array}{lll} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & c \end{array}\right] \end{array} $$

Use the inverse matrices to find (a) \((A B)^{-1},(b)\left(A^{T}\right)^{-1},\) and \((c)(2 A)^{-1}\) $$A^{-1}=\left[\begin{array}{rrr}1 & -\frac{1}{2} & \frac{3}{4} \\\\\frac{3}{2} & \frac{1}{2} & -2 \\\\\frac{1}{4} & 1 & \frac{1}{2}\end{array}\right], \quad B^{-1}=\left[\begin{array}{rrr}2 & 4 & \frac{5}{2} \\\\-\frac{3}{4} & 2 & \frac{1}{4} \\\\\frac{1}{4} & \frac{1}{2} & 2\end{array}\right]$$

Verify that \((A B)^{T}=B^{T} A^{T}\). $$ A=\left[\begin{array}{rrr}2 & 1 & -1 \\\0 & 1 & 3 \\\4 & 0 & 2\end{array}\right] \text { and } B=\left[\begin{array}{rrr}1 & 0 & -1 \\ 2 & 1 & -2 \\\0 & 1 & 3\end{array}\right] $$

Find the steady state matrix \(X\) of the absorbing Markov chain with matrix of transition probabilities \(P\). $$P=\left[\begin{array}{llll} 0.7 & 0 & 0.2 & 0.1 \\ 0.1 & 1 & 0.5 & 0.6 \\ 0 & 0 & 0.2 & 0.2 \\ 0.2 & 0 & 0.1 & 0.1 \end{array}\right]$$

Explain how you can determine the steady state matrix \(X\) of an absorbing Markov chain by inspection.

Prove that the product of two \(2 \times 2\) stochastic matrices is stochastic.

Find \(x\) such that the matrix is equal to its own inverse. $$A=\left[\begin{array}{rr}2 & x \\\\-1 & -2\end{array}\right]$$

Determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. (a) Matrix addition is commutative. (b) The transpose of the product of two matrices equals the product of their transposes; that is, \((A B)^{T}=A^{T} B^{T}\). (c) For any matrix \(C\) the matrix \(C C^{T}\) is symmetric.

Prove that if \(A\) and \(B\) are idempotent and \(A B=B A,\) then \(A B\) is idempotent.

Prove that when \(P\) is a regular stochastic matrix, the corresponding regular Markov chain \(P X_{0}, P^{2} X_{0}, P^{3} X_{0}, \ldots\) approaches a unique steady state matrix \(\bar{X}\).