Linear Algebra: Proving AB Not Invertible for mXn Matrix

In summary, if A is an mXn matrix, B is an nXm matrix, and n<m, then AB is not invertible. This can be shown by finding a nonzero vector x such that Bx=0, which would make AB not linearly independent and therefore not invertible. Additionally, when reduced to echelon form, B will have a free variable due to n<m, leading to more than just the trivial solution. This creates a problem for AB being invertible because if AB has an inverse, then for every x, (AB)^(-1)*ABx=x. However, if ABx=0 and x is not zero, this equation cannot hold true.
  • #1
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Homework Statement


If A is an mXn matrix, B is an nXm matirx, and n<m, then AB is not invertible.


Homework Equations





The Attempt at a Solution


By doing A is a 2X1 and B is a 1X2, I find that AB is not linearly independent, so it cannot be invertible, but I'm not sure how to show that for all matrices of this nature.
 
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  • #2
Can you show there is a nonzero vector x such that Bx=0? That would make big problems for AB being invertible. And don't PM people about problems, ok? Just post it on the forums and wait a bit.
 
  • #3
Since n<m, there will be a free variable in the nXm matrix B when reduced to echelon form, correct? So then there is obviously more than the trivial solution.
I'm still confused as to why that creates a problem for AB being invertible.
 
  • #4
If AB has an inverse (AB)^(-1), then for every x, (AB)^(-1)*ABx=x. What happens if ABx=0 and x is not zero?
 

1. What is a mXn matrix?

A mXn matrix is a rectangular array of numbers with m rows and n columns. It is used to represent linear equations and transformations in linear algebra.

2. What does it mean for a matrix to be invertible?

A matrix is considered invertible if it has an inverse, which is another matrix that when multiplied together with the original matrix, produces the identity matrix. In other words, an invertible matrix has a unique solution for every system of linear equations represented by the matrix.

3. How do you prove that AB is not invertible for a mXn matrix?

To prove that AB is not invertible for a mXn matrix, you can show that the determinant of AB is equal to 0. This means that there is no unique solution for the system of linear equations represented by the matrix, making it not invertible.

4. What is the significance of proving that AB is not invertible for a mXn matrix?

Proving that AB is not invertible for a mXn matrix is important because it helps determine if a system of linear equations has a unique solution or not. It also allows us to understand the properties and characteristics of the matrix, which can help us solve other problems in linear algebra.

5. Are there any other methods for proving that AB is not invertible for a mXn matrix?

Yes, there are other methods for proving that AB is not invertible for a mXn matrix. Another approach is to show that the columns of AB are linearly dependent, meaning that one column can be expressed as a linear combination of the other columns. This would result in a determinant of 0 and prove that the matrix is not invertible.

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