Questions: Is [3/5 1; 2/5 0.4; 2/5 -0.1] a stochastic matrix?
First, identify the properties of a stochastic matrix. Select all that apply.
A. The matrix must have more columns than rows.
B. The matrix must have more rows than columns.
C. The sum of the entries in each column is 1 .
D. The matrix must be square.
E. All entries must be greater than or equal to 0 .
F. The sum of the entries in each row is 1.
Transcript text: Is $\left[\begin{array}{cc}\frac{3}{5} & 1 \\ \frac{2}{5} & 0.4 \\ \frac{2}{5} & -0.1\end{array}\right]$ a stochastic matrix?
First, identify the properties of a stochastic matrix. Select all that apply.
A. The matrix must have more columns than rows.
B. The matrix must have more rows than columns.
C. The sum of the entries in each column is 1 .
D. The matrix must be square.
E. All entries must be greater than or equal to 0 .
F. The sum of the entries in each row is 1.
Solution
Solution Steps
To determine if a given matrix is a stochastic matrix, we need to check specific properties. A stochastic matrix must have non-negative entries, and the sum of the entries in each row must be equal to 1. The matrix does not need to be square, so we will verify these conditions for the given matrix.
Step 1: Check Non-Negativity
The entries of the matrix are:
\[
\begin{bmatrix}
0.6 & 1 \\
0.4 & 0.4 \\
0.4 & -0.1
\end{bmatrix}
\]
To determine if the matrix is stochastic, we first check if all entries are non-negative. The entry \(-0.1\) is negative, which means the condition of non-negativity is not satisfied.
Step 2: Check Row Sums
Next, we calculate the sum of each row:
Row 1: \(0.6 + 1 = 1.6\)
Row 2: \(0.4 + 0.4 = 0.8\)
Row 3: \(0.4 - 0.1 = 0.3\)
None of the rows sum to \(1\). Therefore, the condition that the sum of the entries in each row must equal \(1\) is also not satisfied.
Final Answer
Since the matrix fails both conditions of being non-negative and having row sums equal to \(1\), it is not a stochastic matrix.
Thus, the answer is:
\[
\boxed{\text{The matrix is not stochastic.}}
\]