Proving a Matrix to be nonsingular

[B18]

Homework Statement

Prove that if A is an n x n diagonal matrix with nonzero main diagonal elements; that is a(subscript)ii≠0 for all
1 ≤ i ≤ n, then A is nonsingular and find A(superscript)-1 (A inverse)

Homework Equations

AB=BA=I(subscript)n

The Attempt at a Solution

I first started out by stating that givens, such as A is an n x n diagonal matrix with nonzero diagonal elements. I then let B be an n x n matrix, and also stated that B is the inverse of A.

Am I on the right track here? Can I prove that AB=I(sub)n or BA=I(sub)n??

 

[Brian T]

What allows us to know whether a matrix has an inverse i.e. is nonsingular?

Figuring out B should be easier than the proof itself.

When you multiply two diagonal matrices, in this case A and B, the product is a diagonal matrix where each entry is the product of the corresponding entries.

Use this and the fact that AB = I to determine B

 

[B18]

The matrix must be square in order to have an inverse.
There must exist an n x n matrix B such that AB=BA=I_n.

 

[Brian T]

You're on the right track. It has to be square and something else

For ex, a 2x2 square matrix of all zeros doesn't have a inverse. Why?

 

[B18]

If the matrix A has an inverse that inverse is unique.

 

[Brian T]

But how do we know it has an inverse in the first place? If I gave you a matrix
[2 3]
[1 4]
How do you get its inverse?

 

[B18]

If that is A: AB=I_n.
[2 3][b₁₁ b₁₂] = [1 0]
[1 4][b₂₁ b₂₂] [0 1]

Then just multiply AB and set those equal to the identity matrix.

 

[Mark44]

B18, you're not using a very important piece of the given information -- your matrix A is diagonal.

I then let B be an n x n matrix, and also stated that B is the inverse of A.

You can't assume that B is the inverse of A; i.e., that B = A^-1. You have to first show that A has an inverse; that is, is nonsingular. There is an important tool that can be used on square matrices to determine whether they are singular or nonsingular.

 

[Brian T]

If that is A: AB=I_n.
[2 3][b₁₁ b₁₂] = [1 0]
[1 4][b₂₁ b₂₂] [0 1]

Then just multiply AB and set those equal to the identity matrix.

B18, you're not using a very important piece of the given information -- your matrix A is diagonal.

You can't assume that B is the inverse of A; i.e., that B = A^-1. You have to first show that A has an inverse; that is, is nonsingular. There is an important tool that can be used on square matrices to determine whether they are singular or nonsingular.

As Mark said, you cannot assume that A has an inverse matrix and then set out on finding it. You need a way to see if it has an inverse before you can attempt to use your method to solve that example problem I gave you.

 

[B18]

For a square matrix to be nonsingular the determinant cannot equal 0.

 

[Mark44]

For a square matrix to be nonsingular the determinant cannot equal 0.

Bingo!

And calculating the determinant of a diagonal matrix is very easy. For your matrix, a_ii ≠ 0, for i = 1, ..., n, so what is |A|?

 

[Ray Vickson]

As Mark said, you cannot assume that A has an inverse matrix and then set out on finding it. You need a way to see if it has an inverse before you can attempt to use your method to solve that example problem I gave you.

On the other hand, if you have some given matrix A and I hand you a matrix B and ask you if B is the inverse of A, you don't need to know if A has an inverse; you just need to check if AB = I. That could be the situation if I started blindly using a matrix-inversion algorithm on your A and it went all the way to the end without encountering any problems.

 

[B18]

Well A should be: a₁₁a₂₂ if it were a 2x2 matrix.

 

[Mark44]

Well A should be: a₁₁a₂₂ if it were a 2x2 matrix.

Do you mean |A| -- the determinant of A? In any case, the given matrix is n x n.

 

[B18]

Yes the determinant of A would be that. So in my proof I need to state that the determinant of A is not zero because it has nonzero diagonal values. Correct?

 

[Mark44]

Yes the determinant of A would be that. So in my proof I need to state that the determinant of A is not zero because it has nonzero diagonal values. Correct?

Instead of just stating that |A| ≠ 0, your proof would be more convincing if you showed that it was not zero. In your class have you spent any time on determinant properties and theorems?

 

[B18]

Mark- We have not covered the determinant properties or theorems yet. They are in the next chapter. The chapter that this proof is from contains a small part on how the determinant cannot be 0 if the matrix has an inverse.

 

[B18]

Am I going to need to write out the A and B matrices in order to prove this? All the proofs in this section thus far have been denoted by A,B, and C. I haven't yet done a proof where i needed to write out the matrix and fill in the entries.

 

[Mark44]

Mark- We have not covered the determinant properties or theorems yet. They are in the next chapter. The chapter that this proof is from contains a small part on how the determinant cannot be 0 if the matrix has an inverse.

What exactly does that part say? As you gave it here, it's of no use, since it starts with "if a matrix has an inverse" and you don't know that.

 

[Mark44]

Am I going to need to write out the A and B matrices in order to prove this?

There is no matrix B given in this problem. All you have to go on is an n X n diagonal matrix A.

All the proofs in this section thus far have been denoted by A,B, and C. I haven't yet done a proof where i needed to write out the matrix and fill in the entries.

 

[B18]

If A is a 2 x 2 matrix given by
A=[a b]
...[c d]
then A is invertible if and only if ad-bc ≠ 0. Moreover, if ad-bc≠0, then the

inverse is given by
1/(ad-bc) * [a b]
... [d c]

 

[Mark44]

If A is a 2 x 2 matrix given by
A=[a b]
...[c d]
then A is invertible if and only if ad-bc ≠ 0. Moreover, if ad-bc≠0, then the

inverse is given by
1/(ad-bc) * [a b]
... [d c]

None of this is relevant. The matrix in your problem is n x n and diagonal.

You said there was a short section on determinants in your book. What exactly does it say?

 

[B18]

Mark it is a small comment the book makes about the determinant of A being ad-bc, and how this topic will be covered later in the text.

However I found this- adjoining the matrix A with the identity matrix I: [ A I ] then use elementary row options to get the form to [ I A^-1 ]
"The process shown in Example 3 applies to any n x n matrix A and will find the inverse of A, if it exists." - From the book in regards to the method above.

This is called "finding the inverse of a matrix by Gauss-Jordan Elimination". I believe this is the way I need to prove this. All other theorems in my textbook involve bringing in a second matrix B (which is assumed to be the inverse of A).

 

[Mark44]

What does your book say about diagonal matrices?

 

[B18]

I looked in the index of the book and it only gave two pages for diagonal matrices. One was a homework problem with a diagonal matrix. The other page was in the determinants chapter explaining the determinant of a diagonal matrix is a₁₁a₂₂a₃₃...a_nn

 

[Mark44]

I guess this is the way you need to go...

However I found this- adjoining the matrix A with the identity matrix I: [ A I ] then use elementary row options to get the form to [ I A-1 ]
"The process shown in Example 3 applies to any n x n matrix A and will find the inverse of A, if it exists." - From the book in regards to the method above.

This is called "finding the inverse of a matrix by Gauss-Jordan Elimination". I believe this is the way I need to prove this. All other theorems in my textbook involve bringing in a second matrix B (which is assumed to be the inverse of A).

 

[B18]

Im going to go with that method. Thank you very much for your help Mark, and everyone else. Greatly appreciated.