Find the inverse of the matrix \left[\begin{array}{ccc}9&-2\\-10&9\\\end{array}\right] , if it exist.

Question

Find the inverse of the matrix
`\left[\begin{array}{ccc}9&-2\\-10&9\\\end{array}\right]` , if it exist.

Answer 1

The answer is (b)

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}(d)/(ad-bc) &(-b)/(ad-bc)\\(-c)/(ad-bc) &(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
`(1)/(61)\left[\begin{array}{ccc}9&2\\10&9\end{array}\right]`=

`\left[\begin{array}{ccc}(9)/(61)&(2)/(61)\\(10)/(61) &(9)/(61)\end{array}\right]`

* The answer is (b)