Question

If \[ A= \begin{bmatrix} 1& 7& 49\\ 2& 16& 130\\ 2& 18& 170 \end{bmatrix}, \] then $\det(A)=$

• A. $2^3$
• B. $2^4$
• C. $2^5$
• D. $2^6$

Hint:

Before expanding a determinant, always try elementary row operations to create zeros. This reduces calculations significantly and minimizes mistakes.

Answer 1

The Correct Option is C

Concept:
The determinant of a $3\times3$ matrix can be simplified efficiently using elementary row operations that preserve or simplify determinant evaluation. 

Step 1: Apply row operations.
Given \[ A= \begin{bmatrix} 1 & 7 & 49 \\ 2 & 16 & 130 \\ 2 & 18 & 170 \end{bmatrix}. \] Perform \[ R_2 \to R_2 - 2R_1, \quad R_3 \to R_3 - 2R_1. \] Then \[ \begin{bmatrix} 1 & 7 & 49 \\ 0 & 2 & 32 \\ 0 & 4 & 72 \end{bmatrix}. \] 

Step 2: Expand along the first column.
\[ \det(A) = \begin{vmatrix} 2 & 32 \\ 4 & 72 \end{vmatrix} \] \[ = 2(72) - 32(4) \] \[ = 144 - 128 \] \[ = 16 \] 

Step 3: Express as a power of 2.
\[ 16 = 2^4 \] Therefore, \[ \boxed{2^4} \]