Can a 3x3 Upper Triangular Matrix Be Symmetric?

[tonic16]

Homework Statement

Give an example of the matrix:
3x3 upper triangular symmetric matrix

Homework Equations

The Attempt at a Solution

I know what an upper triangular matrix and what a symmetric matrix looks like. But what happens when they put it together? Is the symmetry in the upper right portion of the matrix now? Like this?

1 5 3
0 2 5
0 0 3

 

[Mark44]

Homework Statement

Give an example of the matrix:
3x3 upper triangular symmetric matrix

Homework Equations

The Attempt at a Solution

I know what an upper triangular matrix and what a symmetric matrix looks like. But what happens when they put it together? Is the symmetry in the upper right portion of the matrix now? Like this?

1 5 3
0 2 5
0 0 3

What is the definition of a symmetric matrix?

 

[tonic16]

Numbers are equal across the diagonal.

 

[Mark44]

Numbers are equal across the diagonal.

Which diagonal is that?

 

[tonic16]

Middle diagonal like this:

1 2 4
2 3 7
4 7 5

 

[Mark44]

Yes, and that's often called the main diagonal. So if a 3x3 matrix is symmetric and upper triangular, then a₁₂ has to equal a₂₁, for example. Since it is upper triangular, what must a₂₁ equal?

 

[tonic16]

So you are implying that the symmetric in this this case wouldn't even matter at all since there is upper triangular in it?

Answer should be something like this?

1 2 4
0 3 7
0 0 5

Same answer if the problem asked for an upper triangular example?

 

[Mark44]

No, I am not saying that. Your matrix in post #7 is not symmetric, because a₁₂ = 2, while a₂₁ = 0.

Maybe you have a flawed understanding of what "upper triangular" means. How do you define this term?

 

[tonic16]

Upper triangular would have zeros everywhere southwest of the main diagonal

 

[Mark44]

And if such a matrix is also symmetric, what can you conclude?