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Unformatted text preview: Solutions to even problems from section 2.4 6. Suppose bracketleftbigg X Y Z bracketrightbigg bracketleftbigg A B C bracketrightbigg = bracketleftbigg I 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 Y Z bracketrightbiggbracketleftbigg A B C bracketrightbigg = bracketleftbigg XA 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 I bracketrightbigg bracketleftbigg X Y Z I bracketrightbigg = bracketleftbigg I 0 0 0 0 I bracketrightbigg...
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