Matrix Multiplication and Inverses

In summary, the problem asks to show that A and Inverse(I+A) commute, with the assumption that the inverse of A exists. The solution is shown using the property of inverse matrices and the fact that (I+A)*(I+A)^(-1)=(I+A)^(-1)*(I+A)=I. The existence of A^(-1) is not necessary to prove this result.
  • #1
Random Variable
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Homework Statement



Show that A and Inverse(I+A) commute (where I is the identity matrix).


Homework Equations



Inverse(Inverse(A))=A

Inverse(AB)=Inverse(B)*Inverse(A)


The Attempt at a Solution



My solution assumes the existence of the inverse of A.

A*Inverse(I+A) = Inverse(Inverse(A))*Inverse(I+A)
= Inverse[(I+A)*Inverse(A)]
= Inverse[Inverse(A)+I]
= Inverse[Inverse(A)*(I+A)]
= Inverse(I+A)*Inverse(Inverse(A))
= Inverse(I+A)*A

My professor told me that the inverse of A may or may not exist. Does he want me to prove that it does exist? Can you even prove it does exist from the fact that the inverse of (I+A) exists? Does he want a different proof? Is he just giving me a hard time?
 
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  • #2
You can't prove A^(-1) exists. It might not. Suppose A=0. The result is still true. If (I+A)^(-1) exists then you must have (I+A)*(I+A)^(-1)=(I+A)^(-1)*(I+A)=I. Isn't that enough to prove it without assuming A^(-1) exists?
 
  • #3
Yes, that's all you need. Thanks.
 

Related to Matrix Multiplication and Inverses

What is matrix multiplication?

Matrix multiplication is a mathematical operation that involves multiplying two matrices to create a new matrix. It is represented by the symbol "x" or by simply placing the matrices next to each other. Matrix multiplication follows specific rules and can only be performed on matrices where the number of columns in the first matrix is equal to the number of rows in the second matrix.

What is the purpose of matrix multiplication?

The purpose of matrix multiplication is to combine the elements of two matrices to create a new matrix that represents a transformation or relationship between the two original matrices. It is commonly used in fields such as physics, engineering, and computer science to solve systems of equations, perform transformations, and analyze data.

What is a matrix inverse?

A matrix inverse is a matrix that, when multiplied by the original matrix, results in the identity matrix (a matrix with 1s on the main diagonal and 0s everywhere else). It is denoted by the exponent "-1" and is used to "undo" a matrix operation, similar to how taking the reciprocal of a number undoes multiplication by that number.

Why is finding the inverse of a matrix important?

Finding the inverse of a matrix is important because it allows us to solve systems of equations, perform transformations, and analyze data more efficiently. It also helps in solving problems where division is involved, as the inverse of a matrix is used to "divide" by a matrix.

How do you find the inverse of a matrix?

To find the inverse of a matrix, we use a specific formula known as the "matrix inverse formula." This involves finding the determinant of the matrix, calculating the matrix of cofactors, and then transposing the resulting matrix. However, for larger matrices, it is more efficient to use a calculator or software program to find the inverse.

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