The value of the determinant \(\begin{vmatrix} 5^2 & 5^3 & 5^4 \\ 5^3 & 5^4 & 5^5 \\ 5^4 & 5^5 & 5^6 \end{vmatrix}\) is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 0
Each element of the second row is 5 times the element above it in the first row:
\(5^3 = 5 \times 5^2,\quad 5^4 = 5 \times 5^3,\quad 5^5 = 5 \times 5^4\)
Taking the common factor 5 out of \(R_2\):
\( \displaystyle \Delta = 5\begin{vmatrix} 5^2 & 5^3 & 5^4 \\ 5^2 & 5^3 & 5^4 \\ 5^4 & 5^5 & 5^6 \end{vmatrix} \)
Now \(R_1\) and \(R_2\) are the same, and a determinant with two identical rows is zero. So \(\Delta = 0\).