Question

How do you find the determinant of `((1, 2, 3), (4, 5, 6), (7, 8, 9))`?

Answer 1

The determinant is `0`.

Explanation:

Since column 3 of the given matrix consists of numbers which are all constant multiples of each other, there is a theorem which states that the determinant of this matrix is `0`.
However, I will prove it from calculation just to verify and show how you would calculate such determinants should you not be aware of the theorems of linear matrix algebra.

Use co-factor expansion along any row or column of your choice.

This involves adding the products of the entries in each row with their co-factors `(-1)^(i+j)`, for an entry in row I and column j, multiplied by the minor of the entry, formed by evaluating the resulting determinant when you delete row I and column j.
In this case, since the given matrix is `3xx3` in dimension, it will result in three `2xx2` determinants which we can find from definition
`|(a,b),(c,d)|=ad-bc`

I will take co-factor expansion along row 1 as an example to find the determinant of the given matrix as :

`Delta=1(-1)^(1+1)|(5,6),(8,9)|+2(-1)^(1+2)|(4,6),(7,9)|+3(-1)^(1+3)|(4,5),(7,8)|`

`=1(45-48)+(-2)(36-42)+3(32-35)`

`=-3+12-9`

`=0`