The group of invertible 2 by 2 matrices

In summary, The subgroups of GL(2,R) are T = {All A in GL(2,R) for which det A = 2}, U = {All A in GL(2,R) for which A is upper triangular}, and V = {All A in GL(2,R) for which trace(A) = 0}. The only correct subgroup is II, which is closed under multiplication, has the identity, and has an inverse. I and III are not correct because I is not closed under multiplication and III does not have an identity. The unit matrix in GL(2,R) has a determinant of 1 and a trace of 2. The axioms for a group include closure, associativity, identity,
  • #1
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This is my third day studying number theory and abstract algebra, and I'm confused about the concept of subgroups.

Which one or more of the following are subgroups of GL(2,R), the group of invertible 2 by 2 matrices with real entries under matrix multiplication?

I. T = {All A in GL(2,R) for which det A = 2}
II. U = {All A in GL(2,R) for which A is upper triangular.}
III. V = {All A in GL(2,R) for which trace(A) = 0}

The answer is II only.

My textbook has an explanation, but does not go into too much detail.

What are ALL the reasons for which the I. and III. are wrong and II. is correct? E.g., which subgroups are closed under multiplication, have the identity, and have an inverse?
 
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  • #2
Actually, it doesn't make much sense to ask about inverses when there is no identity.

That said, if you look at I, you will discover that I is not closed over multiplication.

For III, you can see that the trace of the identity matrix is non-zero, so there is no identity, even if the set is closed under multiplication.
 
  • #3
What determinant and trace does the unit matrix in GL(2,R) have?

What are the axioms for a group?
 

1. What is a 2 by 2 matrix?

A 2 by 2 matrix is a rectangular array of numbers or variables arranged in two rows and two columns. It is represented as A = [a11 a12; a21 a22], where aij represents the element in the i-th row and j-th column.

2. What does "invertible" mean in the context of matrices?

Invertible refers to the property of a matrix to have an inverse, which is another matrix that when multiplied with the original matrix, gives the identity matrix as the result. In other words, an invertible matrix has a unique solution for the equation Ax = b, where A is the matrix, x is a vector of variables, and b is a vector of constants.

3. How do you determine if a 2 by 2 matrix is invertible?

A 2 by 2 matrix is invertible if its determinant, denoted as |A| or det(A), is non-zero. The determinant of a 2 by 2 matrix A = [a11 a12; a21 a22] is given by |A| = a11a22 - a12a21.

4. What is the significance of the group of invertible 2 by 2 matrices?

The group of invertible 2 by 2 matrices, denoted as GL(2), is a mathematical group that has important applications in linear algebra and geometry. It represents all the possible transformations that can be applied to a 2-dimensional vector space, such as rotations, reflections, and shears.

5. Can a 2 by 2 matrix be invertible if it has only zero entries?

No, a 2 by 2 matrix cannot be invertible if it has only zero entries because its determinant would be zero. Invertibility requires the matrix to have a non-zero determinant, and in this case, the inverse matrix would not exist. However, a matrix with only zero entries is known as a singular matrix and has its own significance in linear algebra.

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