Properties of Determinants
General
Grade 12
Question:
If $\begin{vmatrix} 3^2+k & 4^2 & 3^2+3+k \\ 4^2+k & 5^2 & 4^2+4+k \\ 5^2+k & 6^2 & 5^2+5+k \end{vmatrix} = 0$, then the value of $k$ is
Step-by-Step Solution
Key Concept: General
Applying $(C_3 \rightarrow C_3 - C_1)$<br>$D = \begin{vmatrix} 3^2+k & 4^2 & 3 \\ 4^2+k & 5^2 & 4 \\ 5^2+k & 6^2 & 5 \end{vmatrix} = 0$<br>$\Rightarrow \begin{vmatrix} 9+k & 16 & 3 \\ 7 & 9 & 1 \\ 9 & 11 & 1 \end{vmatrix} = 0$ $(R_3 \rightarrow R_3 - R_2; R_2 \rightarrow R_2 - R_1)$<br>$\Rightarrow k - 1 = 0 \Rightarrow k = 1$
Correct Answer: 1