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Question

If 3A=122212x2y and AAT=I, then find the value of x+y. [3 Marks]

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Solution

Given that,3A=122212x2y
A=⎢ ⎢ ⎢132323231323x323y3⎥ ⎥ ⎥ [0.5 Mark]
AT=⎢ ⎢ ⎢1323x32313232323y3⎥ ⎥ ⎥ [0.5 Mark]
AAT=I⎢ ⎢ ⎢132323231323x323y3⎥ ⎥ ⎥⎢ ⎢ ⎢1323x32313232323y3⎥ ⎥ ⎥=100010001⎢ ⎢ ⎢ ⎢19+49+4929+2949x9+49+2y929+294949+19+492x9+292y9x9+49+2y92x9+292y9x29+49+y29⎥ ⎥ ⎥ ⎥=100010001x2+y2+4=9x2+y2=5 ...(1)

2x+22y=0y=x+1 ...(2)

x+4+2y=0x+2y=4 ...(3)

Solving Equations (1) and (2)

x2+(x+1)2=5x2+x2+2x+1=5x2+x2=0x=2,1(x,y)(1,2),(2,1)

Satisfying these in Equation (3)

(2)+2(1)=4(1)+2(2)4
Hence, (2,1) satisfies.

So x+y=3. [2 Marks]

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