Matrices & Determinants
Matrix multiplication
Grade Class 12
Question:
If $M = \begin{bmatrix} 0 & 2 \\ 5 & 0 \end{bmatrix}$ and $N = \begin{bmatrix} 0 & 5 \\ 2 & 0 \end{bmatrix}$, then $M^{2011}$ is -
$10^{1005} M$
$10^{1005} N$
$10^{2010} M$
$10^{2011} M$
Step-by-Step Solution
Key Concept: Calculate M^2 to find a pattern. M^2 = [0 2; 5 0][0 2; 5 0] = [10 0; 0 10] = 10I. Then M^2011 = M * (M^2)^1005 = M * (10I)^1005 = 10^1005 * M.
Given $M = \begin{bmatrix} 0 & 2 \\ 5 & 0 \end{bmatrix}$. <br> $M^2 = \begin{bmatrix} 0 & 2 \\ 5 & 0 \end{bmatrix} \begin{bmatrix} 0 & 2 \\ 5 & 0 \end{bmatrix} = \begin{bmatrix} 10 & 0 \\ 0 & 10 \end{bmatrix} = 10I$. <br> $M^{2011} = M^{2010} \cdot M = (M^2)^{1005} \cdot M = (10I)^{1005} \cdot M = 10^{1005} I \cdot M = 10^{1005} M$.
Correct Answer: 1