Matrices & Determinants
Matrix multiplication
Grade Class 12
Question:
If P = \begin{pmatrix} 1 & 0 \\ 1/2 & 1 \end{pmatrix}, then P^{50} is:
\begin{pmatrix} 1 & 0 \\ 25 & 1 \end{pmatrix}
\begin{pmatrix} 1 & 50 \\ 0 & 1 \end{pmatrix}
\begin{pmatrix} 1 & 25 \\ 0 & 1 \end{pmatrix}
\begin{pmatrix} 1 & 0 \\ 50 & 1 \end{pmatrix}
Step-by-Step Solution
Key Concept: Calculate P^2, P^3 to find a pattern for P^n using mathematical induction.
Given P = \begin{pmatrix} 1 & 0 \\ 1/2 & 1 \end{pmatrix}. <br> P^2 = \begin{pmatrix} 1 & 0 \\ 1/2 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 \\ 1/2 & 1 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}. <br> P^3 = P^2 \cdot P = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 \\ 1/2 & 1 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 3/2 & 1 \end{pmatrix}. <br> By induction, P^n = \begin{pmatrix} 1 & 0 \\ n/2 & 1 \end{pmatrix}. <br> For n = 50, P^{50} = \begin{pmatrix} 1 & 0 \\ 50/2 & 1 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 25 & 1 \end{pmatrix}.
Correct Answer: 1