determinant {52 53 54 53 54 55 54 55 56}find the value of determinant - Maths - Determinants

Question

determinant {52 53 54
53 54 55
54 55 56}find the value of
determinant

Priyanka Kedia · Accepted Answer

Let

$Δ = | \begin{matrix} 5^{2} & 5^{3} & 5^{4} \\ 5^{3} & 5^{4} & 5^{5} \\ 5^{4} & 5^{5} & 5^{6} \end{matrix} |$

Take $5^{2}, 5^{3}, 5^{4}$ common from row $R_{1}, R_{2} a n d R_{3}$ respectively, we get,

$Δ = (5^{2}) (5^{3}) (5^{4}) | \begin{matrix} 1 & 5 & 5^{2} \\ 1 & 5 & 5^{2} \\ 1 & 5 & 5^{2} \end{matrix} |$

We know that, if any two rows of a determinant are identical then the determinant vanishes. Thus,

$Δ = (5^{2}) (5^{3}) (5^{4}) \times 0 \Rightarrow Δ = 0$

Hence value of determinant is zero.