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Chapter 2: Problem 13

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

Short Answer

Expert verified
The product of two \(2 \times 2\) stochastic matrices is indeed stochastic, as shown by multiplying the matrices and confirming that each row in the resulting matrix sums to 1.

Step by step solution

01

Introduce the matrices

Let's consider two \(2 \times 2\) stochastic matrices \(A = \left[\begin{array}{cc} a & b \ c & d \end{array} \right]\) and \(B = \left[\begin{array}{cc} e & f \ g & h \end{array} \right]\), where \(a, b, c, d, e, f, g, h\) are non-negative real numbers. Since both \(A\) and \(B\) are stochastic, we have that \(a + b = c + d = e + f = g + h = 1\).
02

Multiply the matrices

The product of \(A\) and \(B\) is a new \(2 \times 2\) matrix \(AB = \left[\begin{array}{cc} ae + bg & af + bh \ ce + dg & cf + dh \end{array} \right]\).
03

Check if the resulting matrix is Stochastic

To verify that \(AB\) is stochastic, we must check if each row sums to 1. For the first row, \( (ae + bg) + (af + bh) = a(e + f) + b(g + h) = a * 1 + b * 1 = a + b = 1 \). Similarly, for the second row, \( (ce + dg) + (cf + dh) = c(e + f) + d(g + h) = c * 1 + d * 1 = c + d = 1 \). Therefore, \(AB\) is indeed stochastic.

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Most popular questions from this chapter

Determine whether the matrix is idempotent. A square matrix \(A\) is idempotent if \(A^{2}=A\) $$\left[\begin{array}{ll} 1 & 0 \\ 0 & 0 \end{array}\right]$$

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