Understanding Inverse Matrices with Powers

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In summary, if a square matrix A is invertible, then (A^2)^-1 = (A^-1)^2 and (A^3)^-1 = (A^-1)^3. This is similar to the property (A^T)^-1 = (A^-1)^T and the matrix is nxn. The notation A^2 means multiplying A by itself. These proofs can be solved in 2-3 lines and the first equation is the most challenging.
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courtrigrad
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If [tex] A = [a_{ij}]^{n\times n} [/tex] is invertible, show that [tex] (A^{2})^{-1} = (A^{-1})^{2} [/tex] and [tex] (A^{3})^{-1} = (A^{-1})^{3} [/tex]

So basicaly we have a square matrix with elements [tex] a_{ij} [/tex]. This looks slightly familar to [tex] (A^{T})^{-1} = (A^{-1})^{T} [/tex]. Are [tex] A^{2} [/tex] and [tex] A^{3} [/tex] meant to be the elements of the matrix raised to those respective powers? Or does it mean that the matrix is [tex] 2\times 2 [/tex] or [tex] 3\times 3 [/tex]?
 
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The matrix is nxn. A^2 means AA.

As 90% of linear algebra proofs, these problems are solvable in 2-3 lines. If you really don't find it, I can start you and you will find it imidiately. Laying the first equation is the hardest.
 
  • #3
thanks. I got it, just wasn't clear about the notation.
 

FAQ: Understanding Inverse Matrices with Powers

What is an invertible matrix?

An invertible matrix is a square matrix that has an inverse matrix, which when multiplied together, result in the identity matrix. The inverse matrix of an invertible matrix is denoted by A-1.

How do you determine if a matrix is invertible?

A matrix is invertible if its determinant is non-zero. The determinant of a matrix is calculated by subtracting the product of the elements in the main diagonal from the product of the elements in the opposite diagonal. If the determinant is zero, the matrix is not invertible.

What are the properties of invertible matrices?

Invertible matrices have the following properties:

  • They have a unique inverse matrix.
  • The product of an invertible matrix and its inverse is equal to the identity matrix.
  • The inverse of an invertible matrix is also invertible.
  • The transpose of an invertible matrix is also invertible.

How do you find the inverse of a matrix?

To find the inverse of a matrix, you can use the Gauss-Jordan elimination method or the adjoint method. In the Gauss-Jordan method, the matrix is manipulated using elementary row operations until it is in its reduced row echelon form. The resulting matrix will be the inverse of the original matrix. In the adjoint method, the inverse is calculated using the transpose and determinant of the original matrix.

What is the significance of invertible matrices in scientific research?

Invertible matrices are important in scientific research as they are used to solve systems of equations, calculate transformations in linear algebra, and represent physical properties of objects in physics. They are also used in data analysis, image processing, and machine learning algorithms.

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