Proving Invertible Matrix Property: kth Power

In summary, an invertible matrix is a square matrix with an inverse that can be found by using a specific formula. The invertible matrix property states that a matrix is invertible if and only if its determinant is non-zero. The formula for calculating the inverse of a matrix involves the determinant and the adjugate, which is found by taking the transpose of the matrix of cofactors. The kth power property for an invertible matrix allows for easy calculation of powers and is important in mathematical proofs and applications.
  • #1
ephemeral1
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Homework Statement



Prove: If A is an invertible matrix and k is a positive integer, then
(A^k)^-1 = (A^-1)(A^-1) ...A^-1=(A^-1)^k

Homework Equations


none


The Attempt at a Solution



I have a hard time proving this. How do I go about doing this? Any help would be great. I really want to understand this. Thank you.
 
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  • #2
Just multiply [itex]A^k[/itex] by [itex](A^{-1})^k[/itex] and show you get the identity matrix.
 

1. What is an invertible matrix?

An invertible matrix is a square matrix that has an inverse, meaning it can be multiplied by another matrix to produce the identity matrix. The inverse of a matrix is denoted as A-1 and is calculated by using a specific formula.

2. How is the invertible matrix property defined?

The invertible matrix property, also known as the non-singularity property, states that a square matrix is invertible if and only if its determinant is non-zero. This means that a matrix must have a non-zero determinant in order to have an inverse.

3. What is the formula for calculating the inverse of a matrix?

The formula for calculating the inverse of a matrix A is A-1 = (1/det(A)) * adj(A), where det(A) is the determinant of A and adj(A) is the adjugate of A. The adjugate of A is found by taking the transpose of the matrix of cofactors of A.

4. How do you prove the kth power property for an invertible matrix?

To prove the kth power property for an invertible matrix A, we need to show that (A-1)k = (Ak)-1. This can be done by using the definition of matrix multiplication and the formula for calculating the inverse of a matrix.

5. What is the significance of the kth power property for an invertible matrix?

The kth power property for an invertible matrix is important because it allows us to easily calculate powers of the matrix and its inverse. This property is also used in various mathematical proofs and applications, such as solving systems of linear equations and finding the inverse of a matrix raised to a power.

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