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Hi Randy,

I have an matrix question as shown in the picture attached I don't really know how to approach. The question I am stuck on is question #52 which uses info from 52. Any help would be appreciated.

many many thanks,

Sam

#52: If X = AX + D, then for part A subtract AX from both side to get X - AX = (I - A)X = D using the distributive law and definition of the 2x2 identit matrix, I. Then multipying both sides on the left by (I-A)^-1 gives

X = [(I - A)^-1]D

For part B, you need to compute (I - A)^-1.The inverse is, technically, the adjoint of the matrix of cofactors divided by the determinant. You should have a formula in your textbook for this. Anyway, I get

(I - A)^-1 =

{( 1 0 ) - ( 3 2 )

( 0 1 ) - ( 0 -1)}^-1 =

(-1/2 -1/2 )

( 0 1/2).

where we have used the definition of the identity matrix

I = ( 1 0 )

( 0 1 ).

This gives

X = [(I - A)^-1]D = ( 0 1 )^T

where the T stands for tranpose (row vector to a column vector, just to keep things neat).

You should multiply [(I - A)^-1][(I - A)] to make sure it equals the 2x2 identity matrix.

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