Sunday, June 5, 2011

3.1 – Row & Column Operations

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.


  1. if i add any row or column with a real number, k is the determinant the same?

  2. Absolutely no. You can actually try it out yourself on the determinants.

  3. You are just incredible . I am a form 3 student right now and I found out about what you went through and what you're doing currently via ReCom forum . I am very interested in Science ( particularly Physics ) and Mathematics right now , for my PMR . After SPM , I plan to further my studies in Physics ( BSc in Physics ) . I am very interested in Physics .

    By the way , is STPM very hard ? I am afraid if I have no other options or scholarships , I may have to endeavor the harsh syllabus of STPM . Is it true that many people say that if you venture into STPM , you have no other time than to study .

    You are an inspiration . If I do take Further Math T when I'm in STPM , I hope you're still active online so I can ask you questions . :)

  4. You probably should comment on the FAQ page. Anyway, STPM, if you want to score well, then is really not for those who score 5A's and below in SPM. It is very hard, but the only way for you to do Physics, is either you study A-levels or STPM. A-levels is really not too easy either. In fact, people who studied STPM do better in University 1st year than A-level students, because we already learnt more than they do.