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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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#### randy patton

##### Expertise

college mathematics, applied math, advanced calculus, complex analysis, linear and abstract algebra, probability theory, signal processing, undergraduate physics, physical oceanography

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26 years as a professional scientist conducting academic quality research on mostly classified projects involving math/physics modeling and simulation, data analysis and signal processing, instrument development; often ocean related

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J. Physical Oceanography, 1984 "A Numerical Model for Low-Frequency Equatorial Dynamics", with M. Cane

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M.S. MIT Physical Oceanography, B.S. UC Berkeley Applied Math

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