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Question
Matrices A, B, and C are defined by

A= |5 1|  B= |2 4 |  C=|9 -7|
  |7 2|     |-3 5|    |8  2|

Let X be an unknown 2x2 matrix satisfying the equation AX+B=C.

This equation may be solved for X by rewriting it in the form X= A^-1 D where D is a 2x2 matrix.

a) write down A^-1
b) Find D
c) Find X

Answer
a) The inverse of a square matrix is the "adjunct of A divided by the determinant of A" which is also equal to the "transpose of the matrix of cofactors of A". Hopefully, you have a textbook that defines these quantities. Nonetheless, they are easy to calculate for 2x2 matrices. The question says to "write down A^-1, which makes me think you might have been taught a handy shortcut, but for the matrix A, I get

A^-1 = (1/3)(2  -1)
         (-7  5).

You should check this by calculating AA^-1 = I = identity matrix with 1s on the diagonal and 0 on the off-diagonals.

Note that det(A) = 3.

b) D = C- B

= (9-2  -7-4)
   (8+3  2-5)

= (7  -11)
  (11  -3).

c) X = (A^-1)D.

You should be able to multiply 2 matrices to get the answer.

Send me a follow-up if you need more help.

Randy

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randy patton

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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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