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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

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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