(1.) If \( T={ }_{B}^{A}\left(\begin{array}{cc}0.25 & 0.75 \\ 0.3 & \alpha\end{array}\right) \) is a transition probability matrix, then the value of \( \alpha \) is a) \( 0.5 \) b) \( 0.7 \) c) 0 /d) \( 0.6 \) 2. Rank of matrix \( \left(\begin{array}{cccc}1 & -1 & 2 & 0 \\ 0 & 2 & 0 & 3\end{array}\right) \) is a) 1 b) 0 c) 2 d) \( -1 \)
(1.) If \( T={ }_{B}^{A}\left(\begin{array}{cc}0.25 & 0.75 \\ 0.3 & \alpha\end{array}\right) \) is a transition probability matrix, then the value of \( \alpha \) is
a) \( 0.5 \)
b) \( 0.7 \)
c) 0
/d) \( 0.6 \)
2. Rank of matrix \( \left(\begin{array}{cccc}1 & -1 & 2 & 0 \\ 0 & 2 & 0 & 3\end{array}\right) \) is
a) 1
b) 0
c) 2
d) \( -1 \)
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2 Answers
(1) ∵ Row sum of a transition probability matrix must be 1.
∴ 0.3 + α = 1
(sum of probability of 2nd row)
\(\Rightarrow\) α = 1 - 0.3 = 0.7
(2) \( \begin{bmatrix} 1 & -1 & 2&0 \\[0.3em] 0 & 2 & 0&3 \\[0.3em] \end{bmatrix}\)
∴ Rank of matrix is 2.
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