Show that matrices of defined form have inverse of the same same defined form

In summary, the conversation discusses proving that the set of 3x3 matrices of the form [1, a, b; 0, 1, c; 0, 0, 1] with real number entries is a group under matrix multiplication. The main difficulty is showing that the inverses of these matrices also have the same form, which was eventually solved by using elementary row operations.
  • #1
donald17
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Homework Statement



Given the set of 3x3 matrices of the form: [1, a, b; 0, 1, c; 0, 0, 1], where a, b, and c are any real numbers show that the inverses of these matrices are of the same given form.

Homework Equations



Using elementary row operations, transform [A:I] into [I:A-1].
Inverse of a 3x3 matrix

The Attempt at a Solution



This is a subsection of a problem in which I am attempting to show that the set of these 3x3 matrices are a group under matrix multiplication. I was able to prove that it is well-defined, closed, an identity exists, and that associativity holds. For the inverse, it was simple to show that this set of 3x3 matrices is non-singular, but the trouble I'm running into is showing that the inverse is of the same given form so that closure still holds.

Thanks for any assistance.
 
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  • #2
donald17 said:

Homework Statement



Given the set of 3x3 matrices of the form: [1, a, b; 0, 1, c; 0, 0, 1], where a, b, and c are any real numbers show that the inverses of these matrices are of the same given form.

Homework Equations



Using elementary row operations, transform [A:I] into [I:A-1].
Inverse of a 3x3 matrix

The Attempt at a Solution



This is a subsection of a problem in which I am attempting to show that the set of these 3x3 matrices are a group under matrix multiplication. I was able to prove that it is well-defined, closed, an identity exists, and that associativity holds. For the inverse, it was simple to show that this set of 3x3 matrices is non-singular, but the trouble I'm running into is showing that the inverse is of the same given form so that closure still holds.
What trouble are you having? Finding the inverse is straightforward, and yields the inverse in just a few steps. The inverse has the same form.
 
  • #3
Actually I just solved it. Thanks.
 

FAQ: Show that matrices of defined form have inverse of the same same defined form

1. What is the definition of an inverse matrix?

An inverse matrix is a square matrix that, when multiplied by the original matrix, results in the identity matrix. The identity matrix is a square matrix with 1's along the main diagonal and 0's everywhere else.

2. How can you show that a matrix has an inverse of the same form?

To show that a matrix has an inverse of the same form, you can use the determinant of the matrix. If the determinant is non-zero, then the matrix has an inverse of the same form. Additionally, you can use elementary row operations to transform the matrix into the identity matrix, which proves the existence of an inverse of the same form.

3. What are the different forms of matrices that have an inverse?

There are several forms of matrices that have an inverse, including square matrices, diagonal matrices, upper triangular matrices, and lower triangular matrices. These matrices all have the property that their determinants are non-zero, which allows for the existence of an inverse of the same form.

4. Is it possible for a matrix to have an inverse of a different form?

No, it is not possible for a matrix to have an inverse of a different form. The inverse of a matrix must have the same dimensions as the original matrix, and the same form as well. This is necessary for the inverse to cancel out the original matrix when multiplied together.

5. Can any matrix have an inverse of the same form?

No, not every matrix has an inverse of the same form. As mentioned before, the determinant of a matrix must be non-zero for it to have an inverse of the same form. Additionally, singular matrices, or matrices with a determinant of 0, do not have an inverse at all.

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