Matrices & Determinants
Matrices And Determinants
nta_abhyas_2025
Grade 12
Question:
Given $A = \sum_{i=1}^{4} A_i$ where $A_i = \sum_{r=i}^{4} r \cdot C_r$, $\sum_{n=2}^{2} = $ and $\sum_{n=2}^{4} (4 - r) \cdot C_r$. Now, $\sum_{r=1}^{4} r \cdot C_r = 4 \times 2^3 = 32$. Find $|A| = 1024 - 150 = 874$.
Step-by-Step Solution
Key Concept: Use binomial coefficient identities like $\sum_{r=0}^{n} r \cdot \binom{n}{r} = n \cdot 2^{n-1}$ to evaluate determinants of matrices with binomial entries.
We compute $A = \begin{bmatrix} 32 & 10 \\ 15 & 32 \end{bmatrix}$ using binomial coefficient identities. Using the formula $\sum_{r=1}^{n} r \cdot C_r = n \cdot 2^{n-1}$ and applying it to our summations, we calculate $|A| = 32 \times 32 - 10 \times 15 = 1024 - 150 = 874$.
Correct Answer: 874