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2.4soln

# 2.4soln - Solutions to even problems from section 2.4 6...

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Solutions to even problems from section 2.4 6. Suppose bracketleftbigg X 0 Y Z bracketrightbigg bracketleftbigg A 0 B C bracketrightbigg = bracketleftbigg I 0 0 I bracketrightbigg Find formulas for X,Y , and Z in terms of A,B , and C (assume A and C are square). Solution: Since bracketleftbigg X 0 Y Z bracketrightbigg bracketleftbigg A 0 B C bracketrightbigg = bracketleftbigg XA 0 Y A + ZB ZC bracketrightbigg we have the following equations XA = I, Y A + ZB = 0 , ZC = I. Since A and C are square the first and last equations tell us that both A and C are invertible and that X = A - 1 and Z = C - 1 (by the corollary to the inverse matrix theorem in the blue box on page 130). Thus the second equation becomes Y A + C - 1 B = 0. Now (knowing that A and C are invertible) we can solve for Y as follows: Y A + C - 1 B = 0 Y A = - C - 1 B Y AA - 1 = - C - 1 BA - 1 Y = - C - 1 BA - 1 8. Suppose bracketleftbigg A B 0 I bracketrightbigg bracketleftbigg X Y Z 0 0 I bracketrightbigg = bracketleftbigg I 0 0 0 0 I bracketrightbigg Find formulas for X,Y , and Z in terms of A and C (assume A is square).

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