Question

Find the inverse of the matrix %5Cend%7Barray%7D%5Cright%5D" id="TexFormula1" title="\left[\begin{array}{ccc}9&-2\\-10&9\\\end{array}\right]" alt="\left[\begin{array}{ccc}9&-2\\-10&9\\\end{array}\right]" align="absmiddle" class="latex-formula"> , if it exist.

Answer 1

Answer:

The answer is (b)

Step-by-step explanation:

* Lets check how to find the inverse of the matrix,

 its dimensions is 2 × 2

* To know if the inverse of the matrix exist find the determinant

- If its not equal 0, then it exist

* How to find the determinant

- It is the difference between the multiplication of

 the diagonals of the matrix

Ex: If the matrix is $\left[\begin{array}{ccc}a&b\\c&d\end{array}\right]$

     its determinant = ad - bc

- After that lets swap the positions of a and d, put negatives

 in front of b and c, and divide everything by the determinant

- The inverse will be $\left[\begin{array}{ccc}\frac{d}{ad-bc} &\frac{-b}{ad-bc}\\\frac{-c}{ad-bc} &\frac{a}{ad-bc}\end{array}\right]$

* Lets do that with our problem

∵ The determinant = (9 × 9) - (-2 × -10) = 81 - 20 = 61

- The determinant ≠ 0, then the inverse is exist

∴ The inverse is $\frac{1}{61}\left[\begin{array}{ccc}9&2\\10&9\end{array}\right]$=

   $\left[\begin{array}{ccc}\frac{9}{61}&\frac{2}{61}\\\frac{10}{61} &\frac{9}{61}\end{array}\right]$

* The answer is (b)