Question

[tex] \left|\begin{array}{ccc}5²&5³&5⁴\\5¹&5²&5³\\5³&5⁴& 5⁵\end{array}\right|,=.............,Select Proper option from the given options.

(a) 5⁹

(b) 5¹²

(c) 5⁰

(d) 0

[/tex]

Answer 1

Dear Student,

Answer: Option D ( 0) is the answer

Solution:

$\left|\begin{array}{ccc}{5}^{2}&{5}^{3}&{5}^{4}\\{5}^{1}&{5}^{2}&{5}^{3}\\{5}^{3}&{5}^{4}&{5}^{5}\end{array}\right|$

Now as we can see ,

${5}^{2}$ is common in first Row

${5}^{1}$ is common in second Row

${5}^{3}$ is common in third Row

As in Determinant we can take common from rows or columns individually

So,

= ${5}^{2}$(5)${5}^{3}$ $\left|\begin{array}{ccc} 1&5&{5}^{2}\\1&5&{5}^{2}\\1&5&{5}^{2}\end{array}\right|$


As we know that if only two rows are identical then Determinant will be zero, So

$\left|\begin{array}{ccc} 1&5&{5}^{2}\\1&5&{5}^{2}\\1&5&{5}^{2}\end{array}\right|$ =0

so,
= ${5}^{2}$(5)${5}^{3}$ (0)

= 0

Hope it helps you.