Question

Solve!!!!Find the value

Topic: Matrices and determinants


Source : Nda 2021(1)​

Answer 1

$\large\underline{\sf{Solution-}}$

Wᴇ ᴋɴᴏᴡ ᴛʜᴀᴛ,

↝ nᵗʰ term of an geometric sequence is,

$\begin{gathered}\red\bigstar\:\:{\underline{\orange{\boxed{\bf{\green{a_n\:=\:a {r}^{n - 1}}}}}}} \\ \end{gathered}$

Wʜᴇʀᴇ,

• aₙ is the nᵗʰ term.

• a is the first term of the sequence.

• n is the no. of terms.

• r is the common ratio.

Tʜᴜs,

$\rm :\longmapsto\:a_1 = a$

$\rm :\longmapsto\:a_2 = ar$

$\rm :\longmapsto\:a_3 = a {r}^{2}$

$\rm :\longmapsto\:a_4= a {r}^{3}$

.

.

.

.

$\rm :\longmapsto\:a_9= a {r}^{8}$

Now,

Consider,

$\rm :\longmapsto\:loga_{n + 3} \: - \: loga_n$

$\rm \: = \: log\bigg(\dfrac{a_{n + 3}}{a_n} \bigg)$

$\rm \: = \: log\bigg(\dfrac{a {r}^{n + 2} }{a {r}^{n - 1} } \bigg)$

$\rm \: = \: log\bigg(\dfrac{{r}^{n + 2} }{{r}^{n - 1} } \bigg)$

$\rm \: = \: log {r}^{n + 2 - n + 1}$

$\rm \: = \: log \: {r}^{3}$

Hence,

$\rm :\longmapsto\:loga_{n + 3} \: - \: loga_n \: = \: log \: {r}^{3} - - - (1)$

Now,

Consider,

$\rm :\longmapsto\: \begin{gathered}\sf \left | \begin{array}{ccc}loga_1&loga_2&loga_3\\loga_4&loga_5& loga_6\\loga_7& loga_8& loga_9\end{array}\right | \end{gathered}$

$\rm :\longmapsto\:OP \: R_3 \to \: R_3 - R_2$

$\rm \: \: = \: \begin{gathered}\sf \left | \begin{array}{ccc}loga_1&loga_2&loga_3\\loga_4&loga_5& loga_6\\loga_7 - loga_4& loga_8 - loga_5& loga_9 - loga_6\end{array}\right | \end{gathered}$

$\rm :\longmapsto\:OP \: R_2 \to \: R_2 - R_1$

$\rm \: \: = \: \begin{gathered}\sf \left | \begin{array}{ccc}loga_1&loga_2&loga_3\\loga_4 - loga_1&loga_5 - loga_2& loga_6 - loga_3\\loga_7 - loga_4& loga_8 - loga_5& loga_9 - loga_6\end{array}\right | \end{gathered}$

By using equation (1), we get

$\rm \: = \: \: \begin{gathered}\sf \left | \begin{array}{ccc}loga_1&loga_2&loga_3\\log \: {r}^{3}&log \: {r}^{3}& log \: {r}^{3} \\log\: {r}^{3}& log \: {r}^{3}& log\: {r}^{3}\end{array}\right | \end{gathered}$

As second row and third row are identical, so determinant value is 0.

$\rm \: = \: 0$

Hence,

$\red{\rm :\longmapsto\: \begin{gathered}\sf \left | \begin{array}{ccc}loga_1&loga_2&loga_3\\loga_4&loga_5& loga_6\\loga_7& loga_8& loga_9\end{array}\right | \end{gathered} = \bf \: 0}$

Additional Information :-

1. The value of determinant doesn't change if rows or columns are interchanged.

2. The value of determinant is 0, if elements of any row or column, all are 0.

3. The value of determinant is 0, if any two rows or columns are identical.

4. The value of determinant is multiplied by (-1) if successive rows or columns are interchanged.