Matrices & Determinants
Matrices And Determinants
nta_abhyas_2025
Grade 12
Question:
For $x > 0$, let $A = \begin{bmatrix} x+1 & 0 & 0 \\ 0 & \frac{1}{x} & 0 \\ 0 & 0 & 12 \end{bmatrix}$, $B = \begin{bmatrix} 0 & a_2 & a_3 \\ 0 & \frac{x}{3} & 0 \\ 0 & 0 & \frac{3}{x} \end{bmatrix}$ be two matrices and $C = AB + (AB)^2 + \cdots + (AB)^n$. Then $\operatorname{Tr}\left(\lim_{n \to \infty} C\right)$ is equal to
$1$
$\frac{3}{5}$
$\frac{5}{3}$
$\frac{4}{3}$
Step-by-Step Solution
Key Concept: Trace of a matrix is the sum of diagonal elements, and limits can be applied element-wise to matrices
Starting with matrix $AB = \begin{vmatrix} 2 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{vmatrix}$ and $(AB)^2 = \begin{vmatrix} 4 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{vmatrix}$, we construct matrix $C$ with the pattern shown. Computing the limit of $C$ as it approaches the given form and then finding $\text{Tr}(\lim C) = \frac{1 + 1 + 1}{1} = 0$.
Correct Answer: 0