Symmetric, skew-symmetric and orthogonal matrices. Hello friends, today it’s all about symmetric, skew-symmetric and orthogonal matrices.

**Symmetric, skew-symmetric and orthogonal matrices**

Suppose I have a matrix . Then the matrix will be symmetric if the transpose of the matrix is the same as the original matrix.

Now I give an example.

Let’s say I have a matrix like

Now I’ll check if it’s a symmetric matrix.

First of all, I’ll transpose the matrix. And this means the first row will be the first column. Then the second row will be the second column. And it will continue in this way.

So the transpose of the matrix is . Thus the value of will be

Now the matrix will be

and

Okay, now I’ll give five different examples on symmetric, skew-symmetric and orthogonal matrices.

**Solved examples of symmetric, skew-symmetric and orthogonal matrices**

Disclaimer: 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.

Here comes my first example.

** Example 1**

According to Kreyszig (2005)*, “Is the following matrix symmetric, skew-symmetric, or orthogonal?

”

**Solution**

Now here the matrix is

First of all, I’ll give it a name, say . So it will be

Now I’ll check whether it’s a symmetric, skew-symmetric or orthogonal matrix.

**Step 1 **

First of all, I’ll transpose the matrix . So the first row becomes the first column. Then the second row will be the second column. In the end, the third row will be the third column.

Thus the matrix is

As I can see, this is neither nor , that is

So that means the matrix is neither a symmetric or a skew-symmetric matrix.

Next, I’ll check if it’s an orthogonal matrix. Now for that, I’ll get the inverse of the matrix .

**Step 2**

But I won’t use the standard way to get the inverse of the matrix. Instead, I’ll use the Gauss-Jordan method to find out the inverse of the matrix .

As per the Gauss-Jordan method, I’ll include the unit matrix on the right-hand side like

And my aim is to bring the unit matrix on the left-hand side. Also the matrix on the right-hand side will be the inverse of the matrix .

Ok, I’ll start now.

First of all, I’ll interchange row 1 and row 2 to get the equivalent matrix as

Now I’ll add three times row 1 with row 2. And in mathematical terms, it will be

Row 1 + 3 (Row 2).

So this gives the equivalent matrix as

Next, I’ll divide row 2 by four. Simultaneously, I’ll also multiply row 1 with . And in mathematical terms, it will be

Row 1, Row 2.

Hence the equivalent matrix matrix will be

Then I’ll add row 2 to row 1. And in mathematical terms, it will be

Row 1 + Row 2.

So the equivalent matrix is

Now I have already got the unit matrix on the left-hand side. So that means the inverse of the matrix is

As I can see, this is not the matrix , that is

So the conclusion is that the given matrix is not an orthogonal matrix either.

**Conclusion**

As I have seen from Step 1, the given matrix is neither a symmetric or a skew-symmetric one. Also, from Step 2, I have seen that the given matrix is not an orthogonal one either.

So the conclusion is that this specific matrix is none of the symmetric, skew-symmetric or orthogonal matrices. And this is the answer to the given example.

Here comes my second example.

** Example 2**

According to Kreyszig (2005)*, “Is the following matrix symmetric, skew-symmetric, or orthogonal?

”

**Solution**

Now here the matrix is

First of all, I’ll give it a name, say . So it will be

Next, I’ll transpose the matrix . So the first row becomes the first column. Then the second row will be the second column. At the end, the third row will be the third column.

Thus the matrix is

As I can see, this is exactly the matrix , that is

So the conclusion is that the given matrix is a symmetric matrix. And this is the answer to this example.

Now comes my third example on symmetric, skew-symmetric and orthogonal matrices.

##### **Example 3**

According to Kreyszig (2005)*, “Is the following matrix symmetric, skew-symmetric, or orthogonal?

”

**Solution**

Now here the matrix is

First of all, I’ll give it a name, say . So it will be

Next, I’ll transpose the matrix . Again the first row becomes the first column. Then the second row will be the second column. At the end, the third row will be the third column.

Thus the matrix is

Now I can rewrite the matrix as

And this is exactly the matrix . So in this example,

Hence the conclusion is that the given matrix is a skew-symmetric matrix. And this is the answer to this example.

Now comes my next example.

** Example 4**

According to Kreyszig (2005)*, “Is the following matrix symmetric, skew-symmetric, or orthogonal?

”

**Solution**

Now here the matrix is

First of all, I’ll give it a name, say . So it will be

Now I’ll check whether it’s a symmetric, skew-symmetric or orthogonal matrix.

**Step 1 **

First of all, I’ll transpose the matrix . Again the first row becomes the first column. Then the second row will be the second column. In the end, the third row will be the third column.

Thus the matrix is

As I can see, this is neither nor , that is

So that means the matrix is neither a symmetric or a skew-symmetric matrix.

Next, I’ll check if it’s an orthogonal matrix. Now for that, I’ll get the inverse of the matrix .

**Step 2**

But I won’t use the standard way to get the inverse of the matrix. Instead, I’ll use the Gauss-Jordan method to find out the inverse of the matrix .

As per the Gauss-Jordan method, I’ll include the unit matrix on the right-hand side like

And my aim is to bring the unit matrix on the left-hand side. Also the matrix on the right-hand side will be the inverse of the matrix .

Ok, I’ll start now.

First of all, I’ll interchange row 1 with row 3.

Thus the resultant matrix will be

Next, I’ll multiply row 1 with . So this gives

Now I have already got the unit matrix on the left-hand side. So that means the inverse of the matrix is

As I can see, this is exactly the matrix , that is

So the conclusion is that the given matrix is an orthogonal matrix. And this is the answer to this example.

Now comes my last example.

** Example 5**

According to Kreyszig (2005)*, “Is the following matrix symmetric, skew-symmetric, or orthogonal?

”

**Solution**

Now here the matrix is

First of all, I’ll give it a name, say . So it will be

Next, I’ll transpose the matrix . As before, the first row becomes the first column. Then the second row will be the second column. At the end, the third row will be the third column.

Thus the matrix is

As I can see, this is exactly the matrix , that is

So the conclusion is that the given matrix is a symmetric matrix. And this is the answer to this example.

Now here is my last example on symmetric, skew-symmetric and orthogonal matrices.

Dear friends, this is the end of my today’s post on symmetric, skew-symmetric and orthogonal matrices. 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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