Matrices & Determinants
Matrices And Determinants
nta_abhyas_2025
Grade 12
Question:
Let $A = \begin{bmatrix} \frac{1}{2} & -\frac{3}{2} \\ 1 & -\frac{1}{2} \end{bmatrix}$, then the value of sum of all the elements of $A^{100}$ is
Step-by-Step Solution
Key Concept: When $A^2 = I$, then $A^{10} = (A^2)^5 = I^5 = I$
Given $A^2 = \begin{pmatrix} \frac{1}{2} & \frac{u_4}{1} \\ 1 & \frac{u_2}{1} \end{pmatrix} \begin{pmatrix} \frac{1}{2} & \frac{u_4}{1} \\ 1 & \frac{u_2}{1} \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$. We have $A^2 = I$, so $A^{10} = I$ and the sum equals $2$.
Correct Answer: 2