This section is mostly covered in Maths T. So I will only discuss on properties of determinants.
You should possibly know what a determinant is by now, at least for 2 × 2 and 3 × 3 matrices. In Maths T, you are told to evaluate the determinant of a 3 × 3 matrix just by how it is. Here, I’m going to teach you that there is a shortcut operation such that you could calculate the determinant in a faster method.
Sometimes, we are required to change the appearance of the determinant to ease us in our calculations. There are several ways to change the determinants without altering its value:
1. add / subtract any row to any other row
adding the second row to the first row yields:
2. add / subtract any column to any other column
subtracting the first column from the third column yields:
3. add / subtract any multiple of any row / column to any other row / column
adding 3 times of the third row to the first row yields:
4. interchange 2 rows / columns and change its sign (+ / –)
interchanging column 1 and column 2 yields:
5. factor out a constant k from any row / column
factoring out 3 from all 3 rows yield:
Note the difference here between matrix and determinant. You only factor out one ‘3’ if it were a matrix. Don’t make mistakes.
6. transpose the determinant
I think you understand this without illustration, right?
One more interesting fact about determinants is, whenever 2 rows / columns are equal, the value of the determinant is zero.
Knowing how you can simplify the determinants, this gives you the advantage when you calculate inverse matrices. You can now calculate the value of the determinant faster! Besides, it can be useful for situations, just like the one below:
Factorize the following determinant:
So, the answer will be:
Just don’t be careless. The harder part of this chapter has yet to come. ☺