Prove Invertibility of nxn Matrix AB=I

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Then, since AB = I, we have \det (AB) = \det I = 1. Since \det A and \det B are both non-zero, this means that \det A must also be non-zero, which implies that A is invertible. Therefore, every n x n matrix A for which there exists an n x n matrix B such that AB = I must be invertible.
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Homework Statement



Prove that every n x n matrix A for which there exists an n x n matrix B such that AB = I must be invertible. Hint: Use properties of determinants.

Homework Equations



None that I am aware of.

The Attempt at a Solution



I tried finding the inverse of the matrix and multiplying by an elementary matrix. I also tried finding the determinants of a simple matrix and using it's properties but nothing is working :(
 
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sheldonrocks97 said:

Homework Statement



Prove that every n x n matrix A for which there exists an n x n matrix B such that AB = I must be invertible. Hint: Use properties of determinants.

Homework Equations



None that I am aware of.

The Attempt at a Solution



I tried finding the inverse of the matrix and multiplying by an elementary matrix. I also tried finding the determinants of a simple matrix and using it's properties but nothing is working :(

You are supposed to use the fact that [itex]\det (AB) = (\det A) (\det B)[/itex].
 

FAQ: Prove Invertibility of nxn Matrix AB=I

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, the inverse of a matrix "undoes" the original matrix's transformation.

How can I prove the invertibility of a nxn matrix AB=I?

One way to prove the invertibility of a matrix AB=I is by showing that the determinant of the matrix is non-zero. This ensures that the matrix has a unique solution and therefore, an inverse. Another method is by using the row reduction method to show that the matrix can be reduced to the identity matrix.

What are the implications of a matrix being invertible?

An invertible matrix has many important implications in linear algebra. It allows for the use of techniques such as solving systems of equations, finding the inverse of a function, and calculating determinants. It also plays a crucial role in many real-world applications, such as computer graphics, cryptography, and engineering.

Are all nxn matrices invertible?

No, not all nxn matrices are invertible. A matrix must meet certain criteria, such as having a non-zero determinant, to be considered invertible. In fact, matrices with zero determinants are not invertible. Additionally, square matrices with linearly dependent columns or rows are also not invertible.

Can a matrix have more than one inverse?

No, a matrix can only have one inverse. This is because the inverse of a matrix is unique and can be found using specific mathematical operations. If a matrix has more than one inverse, it would contradict this principle.

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