Determinants to solve equations. Hello friends, today I’ll show how to use determinants to solve equations. Have a look!!

If you’re looking for more in determinants, do check out

**How to evaluate and expand determinants**

**How to use determinants to check the consistency of equations**

**Determinants to solve equations**

Here I’ll give some examples to show how to use determinants to solve equations.

**Solved examples of how to use determinants to solve equations **

Disclaimer: None of these examples is 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)*, “Solve the equation

”

**Solution**

Now here the given equation is:

And I have to solve it to get the value of . So I’ll try to bring two s either in the first row or in the first column. Let’s start then.

**Step 1**

First of all, I’ll add columns and to column . And that gives

Now I’ll take out as a common factor. So that means

Next, I’ll subtract row from row . Also, I’ll subtract row from row . Thus it means

As you can see, I have already got two s in the first column. So I don’t need to reduce it further. Now I’ll expand the determinant.

**Step 2**

So I can say

Now I’ll simplify it. Therefore it will be

And that means

So that gives either or . Now this gives

which means

Thus the solutions of this equation are .

Hence I can conclude that this is the answer to the given example.

Now I’ll give another example.

**Example 2**

According to Stroud and Booth (2013)*, “Find the angles between and that satisfy the equation:

”

**Solution**

Now here the given equation is

And I have to solve it to get the values of . So I’ll try to bring two s either in the first row or in the first column. Let’s start then.

**Step 1**

First of all, I’ll add columns and to column . And that gives

Now I’ll simplify the first column. Since , I can say that

So that means

Thus the determinant will be

Now I’ll take out as a common factor. So that means

Next, I’ll subtract row from row . Also, I’ll subtract row from row . Thus it means

As you can see, I have already got two s in the first column. So I don’t need to reduce it further. Now I’ll expand the determinant.

**Step 2**

So I can say

If I simplify it, I’ll get

And this gives

Now is negative when is in between and . Also, for . Thus can only be , if is in between and .

So I can say for ,

or

Hence I have two different values of – one is and the other is .

And that means

or

If

And if

Thus both the angles and satisfy the given equation. Hence I can conclude that this is the answer to the given example.

Dear friends, this is the end of today’s example. Thank you very much for reading this post. Soon I will be back with a new post on a different topic!! Till then, bye, bye.

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