Proving that the matrix is invertible given the eigenvalues?

Thread starter theBEAST
Start date Apr 16, 2012
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Eigenvalues Matrix

In summary, an invertible matrix is a square matrix with a unique inverse that, when multiplied together, results in the identity matrix. To determine if a matrix is invertible, its determinant must be non-zero, which is calculated by taking the product of its eigenvalues. An invertible matrix can have repeated eigenvalues, as long as the determinant is non-zero. The eigenvalues of a matrix determine its invertibility, with all non-zero eigenvalues indicating invertibility and any zero eigenvalues indicating non-invertibility. The size of the matrix can also affect its invertibility, with a larger square matrix requiring more distinct non-zero eigenvalues to be invertible.

Apr 16, 2012

theBEAST

Homework Statement

Given an unknown matrix with eigenvalues 1,2,3, prove that it is invertible?

The Attempt at a Solution

If the det = 0, then there exists an eigenvalue = 0. Since none of the eigenvalues are 0, then the det ≠ 0 and thus the matrix is invertible. Is this a valid proof?

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Apr 16, 2012

HallsofIvy

Science Advisor

Homework Helper

Yes, it is.

Related to Proving that the matrix is invertible given the eigenvalues?

What is the definition of an invertible matrix?

An invertible matrix is a square matrix that has a unique inverse matrix. This means that when the matrix is multiplied by its inverse, it results in the identity matrix.

How can I determine if a matrix is invertible?

A matrix is invertible if its determinant is non-zero. The determinant is calculated by taking the product of the eigenvalues of the matrix. If the determinant is non-zero, then the matrix is invertible.

Can an invertible matrix have repeated eigenvalues?

Yes, an invertible matrix can have repeated eigenvalues. In this case, the matrix will have fewer distinct eigenvalues than its size, but it will still be invertible as long as the determinant is non-zero.

What is the relationship between eigenvalues and invertibility?

The eigenvalues of a matrix determine its invertibility. If all the eigenvalues are non-zero, then the matrix is invertible. If any eigenvalue is zero, the matrix is not invertible.

Does the size of the matrix affect its invertibility?

Yes, the size of the matrix can affect its invertibility. A square matrix of size n x n must have n distinct non-zero eigenvalues to be invertible. This means that the larger the matrix, the more distinct eigenvalues it must have to be invertible.