Matrices & Determinants
Matrices and Determinants
Allen Star Batch
Grade 12
Question:
Consider three matrices $A = \begin{bmatrix} 2 & 1 \\ 4 & 1 \end{bmatrix}$, $B = \begin{bmatrix} 3 & 4 \\ 2 & 3 \end{bmatrix}$ and $C = \begin{bmatrix} 3 & -4 \\ -2 & 3 \end{bmatrix}$, then the value of the sum $tr(A) + tr\left(\frac{ABC}{2}\right) + tr\left(\frac{A(BC)^2}{4}\right) + tr\left(\frac{A(BC)^3}{8}\right) + \ldots \infty$ is
Step-by-Step Solution
Key Concept: The geometric series for the trace formula converges to $tr(2A)$ when $BC = I$ ensures the powers converge.
Given $BC = I$, we use the trace formula: $tr(A) + tr\left(\frac{ABC}{2}\right) + tr\left(\frac{(ABC)^2}{4}\right) + \cdots = tr(A) + tr\left(\frac{A}{2}\right) + tr\left(\frac{A}{4}\right) + \cdots = tr\left(A\left(1 + \frac{1}{2} + \frac{1}{4} + \cdots\right)\right) = tr(2A) = 2tr(A) = 2 × 3 = 6$.
Correct Answer: 6