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Problem 1

# Determine whether the matrix is an absorbing stochastic matrix. $$\left[\begin{array}{ll}\frac{2}{5} & 0 \\ \frac{3}{5} & 1\end{array}\right]$$

Expert verified
The given matrix $$\left[\begin{array}{ll}\frac{2}{5} & 0 \\\ \frac{3}{5} & 1\end{array}\right]$$ is a square matrix that has non-negative elements. However, the sum of each row is not equal to 1 ($$\frac{2}{5}$$ for row 1 and $$\frac{8}{5}$$ for row 2). As it does not fulfill the conditions to be a stochastic matrix, it cannot be an absorbing stochastic matrix.
See the step by step solution

## Step 1: Check if it is a square matrix

The given matrix has 2 rows and 2 columns, so it is a square matrix.

## Step 2: Check if all elements are non-negative

In the given matrix, all elements are non-negative (fractions or whole numbers).

## Step 3: Check if the sum of each row is equal to 1

Compute the sum of each row: - Row 1: $$\frac{2}{5} + 0 = \frac{2}{5}$$ - Row 2: $$\frac{3}{5} + 1 = \frac{8}{5}$$ The sum of Row 1 is not equal to 1, and the sum of Row 2 is also not equal to 1. So the given matrix is not a stochastic matrix. Since it is not a stochastic matrix, there is no need to check further for condition 4. The given matrix does not meet all the conditions; therefore, it is not an absorbing stochastic matrix.

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