Determining the matrix A for the equation in the photo is respected. Thank you in advance

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Find `A` if `(I+2A^(T))^(-1)=[(-1,2),(4,5)]` :

To solve we "undo" the operations that are done to `A` :

`(I+2A^T)^(-1)=[(-1,2),(4,5)]` Take the inverse of both sides:

`I+2A^T=[(-1,2),(4,5)]^(-1)=[(-5/13,2/13),(4/13,1/13)]`

Subtract the 2x2 identity matrix from both sides:

`2A^T=[(-18/13,2/13),(4/13,-12/13)]` Multiply both sides by the scalar `1/2` :

`A^T=[(-9/13,113),(2/13,-6/13)]` Take the transpose of each side:

`A=[(-9/13,2/13),(1/13,-6/13)]` Which is the required matrix.

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(1) Assuming a matrix has an inverse, then `(A^(-1))^(-1)=A`

(2) We can add/subtract properly sized matrices from both sides of an equation and preserve equality.

(3) We can multiply both sides of a matrix equality by a nonzero scalar at any time and preserve equality.

(4) The transpose of the transpose of a matrix is the original matrix -- `(A^T)^T=A`

** To find the inverse of a 2x2 matrix without technology:

Let `A=[(a,b),(c,d)]` be an invertible matrix. (The determinant is nonzero.) Then:

`A^(-1)=1/(ad-bc)[(d,-b),(-c,a)]`

Thank you a lot, my mistake was to develop the inverse in the first part of the equality. Thank's again

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