You can solve for Y from here immediately:Cyborg31 said:Homework Statement
A, B, I-Y^-1, X, Y are n x n invertible matrices
(A(I-Y^-1))^-1 = YB
Solve for Y
Homework Equations
The Attempt at a Solution
(I - Y^-1)^-1A^-1 = YB
(I - Y^-1)^-1A^-1B^-1 = Y
A^-1B^-1 = (I - Y^-1)Y
A^-1B^-1 = Y - I
Don't you mean = (Y- I)(A^-1B^-1)^-1A^-1B^-1*(A^-1B^-1)^-1 = Y - I(A^-1B^-1)^-1
This is wrong now.I = Y - (A^-1B^-1)^-1
Y^-1*I = Y^-1*Y - (A^-1B^-1)^-1
Y^-1 = I - (A^-1B^-1)^-1
Inverse both sides
Y = (I - (A^-1B^-1)^-1)^-1
or is it
Y = I^-1 - A^-1B^-1
Y = I - A^-1B^-1
Is this correct?
An invertible matrix, also known as a non-singular matrix, is a square matrix that has an inverse matrix. This means that it can be multiplied by another matrix to produce the identity matrix. In simpler terms, an invertible matrix is a matrix that can be "undone", or reversed, by another matrix.
To determine if a matrix is invertible, you can use the determinant. If the determinant is non-zero, then the matrix is invertible. Another way is to check if the matrix has a pivot in every row, which indicates that the columns are linearly independent and the matrix is invertible.
The inverse of a matrix is a matrix that when multiplied by the original matrix, results in the identity matrix. The inverse of a matrix is denoted by A-1. In other words, the inverse of a matrix "undoes" the effects of the original matrix.
No, a non-square matrix cannot be invertible. In order for a matrix to have an inverse, it must be a square matrix, meaning it has the same number of rows and columns.
Invertible matrices have many applications in various fields such as engineering, physics, and economics. They are used to solve systems of linear equations, find the inverse of a function, and perform transformations in computer graphics. Invertible matrices also play a crucial role in error correction in data transmission and cryptography.