Invertible 3x3 matrices a subspace of 3x3 matrices

In summary, the conversation is about whether the set of invertible 3x3 matrices is a subspace of 3x3 matrices. The first person believes it is not because the zero matrix is not in the subset, while the second person agrees and adds that it is also not closed under addition. They clarify that the "neutral 0 element" refers to the additive identity.
  • #1
wumple
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Homework Statement



Is the set of invertible 3x3 matrices a subspace of 3x3 matrices?

Homework Equations





The Attempt at a Solution


I think no - the 'neutral 0 element' is not in the subset since the 3x3 0 matrix is not in the subset. Am I right? The book says it's not a subspace because it's not closed under addition, but I'm not sure if my reason is also correct.
 
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  • #2
Your reason is also correct.
 
  • #3
thanks! also, quick question: does the 'neutral 0 element' mean the additive identity?
 
  • #4
wumple said:
thanks! also, quick question: does the 'neutral 0 element' mean the additive identity?

Sure, subspace generally means closed under linear combinations. The zero matrix is the identity.
 
1)

What is an invertible 3x3 matrix?

An invertible 3x3 matrix is a square matrix with a size of 3x3 that has a determinant of non-zero (not equal to 0) and can be multiplied with another 3x3 matrix to produce the identity matrix.

2)

How can I determine if a 3x3 matrix is invertible?

A 3x3 matrix is invertible if its determinant is not equal to 0. The determinant can be calculated by using the diagonal rule or by using the Laplace expansion method.

3)

What is the significance of invertible 3x3 matrices?

Invertible 3x3 matrices are important in linear algebra as they represent a type of matrix that has an inverse, which allows for the solution of systems of linear equations. They are also used in various applications such as computer graphics and engineering.

4)

Is the set of invertible 3x3 matrices a subspace of the set of 3x3 matrices?

Yes, the set of invertible 3x3 matrices is a subspace of the set of 3x3 matrices. This is because it satisfies the three properties of a subspace: it contains the zero vector (identity matrix), it is closed under addition (the sum of two invertible matrices is also invertible), and it is closed under scalar multiplication (multiplying an invertible matrix by a scalar will still result in an invertible matrix).

5)

How can I find the inverse of an invertible 3x3 matrix?

The inverse of an invertible 3x3 matrix can be found by using the adjugate matrix method or by using the Gauss-Jordan elimination method. Both methods involve a series of mathematical operations on the original matrix to produce the inverse.

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