Here I will talk about how to evaluate expand determinants. Of course, to evaluate determinants is a natural thing. Quite understandable!!

But why to bring the issue of expanding it?? Well, the answer is that’s also needed. Have a look at these examples.

#### How to evaluate, expand determinants

In a simple way, the answer is:

In some cases, I need to expand the determinant first. Then only I can evaluate it.

Otherwise, it becomes quite complicated. Or, sometimes it’s even difficult to handle it.

I just give you some examples. That will help!

Note: None of these examples are mine. I have chosen these from some book or books. I have also given the due reference at the end of the post.

So here is the first example.

##### Example 1

According to Stroud and Booth (2013)*, Evaluate:

##### Solution

Here the given determinant is:

Now if I try to evaluate it at this stage, it will look like

Now this already looks difficult to handle.

So just to avoid this, the best way is to expand this determinant.

Now I will use row and column operations of determinants for that.

I will start by subtracting row 3 from row 2.

###### Step 1

So it will look like

Now I will subtract column 3 from both columns 1 and 2. Therefore the determinant will be

At this stage, the numbers in the first row are still large. So I will try to reduce them now.

Next I will subtract row 3 from row 1. Thus the determinant will be

Now I just observe the determinant again.

What I see is that I can still simplify either row 1 or row 3. I will do it on row 1.

Therefore I will subtract row 3 from 1 to get

Now I will work on columns. I will subtract 2 times column 1 from column 2 and column 1 from column 3.

Therefore the determinant will be

At this stage, I can’t expand the determinant further. Now I will evaluate it.

###### Step 2

Therefore it will be

Thus I can conclude that 666 is the value of the determinant. This is the answer to this problem.

Now I will show you another example.

##### Example 2

According to Stroud and Booth (2013)*, ”Factorize:

##### Solution

Here the given determinant is:

Now the task is to factorise it.

In other words, I have to get some common factor or factors out of it.

Like my first example, here also I will start working with columns.

###### Step 1

First of all, I will subtract column 2 from column 1. Thus the determinant will look like

Now I know the standard formula in algebra for cubic expressions. This is

In this case, it will be

Now I can take out as a common factor from column 1 like

###### Step 2

As a second step, I will subtract column 3 from column 2. Therefore the determinant will be

Like before, now also I will use the formula

Therefore the determinant will look like

Now I will take out as a common factor to make it look like

Next I will subtract column 2 from column 1. Thus it will be

I can rewrite as a product of two factors like

Therefore the determinant will be

Now I take out as a common factor from column 1.

The determinant will then look like

I have already reduced column 1 as most as I can. So the next step will be to evaluate the determinant.

###### Step 3

Therefore it will be

Thus my conclusion is that is the factorised form of the determinant.

Dear friends, this is the end of today’s post. Thank you very much for reading this. Please let me know how you feel about it. Soon I will be back again with a new post. Till then, bye, bye!!

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