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Matrices

Question
CBSEENMA12032894

Find ,x, y , a and b if

open square brackets table row cell 2 straight x minus 3 straight y end cell cell straight a minus straight b end cell 3 row 1 cell straight x plus 4 straight y end cell cell 3 straight a plus 4 straight b end cell end table close square brackets equals open square brackets table row 1 cell negative 2 end cell 3 row 1 6 29 end table close square brackets

Solution

We are given that
open square brackets table row cell 2 straight x minus 3 straight y end cell cell straight a minus straight b end cell 3 row 1 cell straight x plus 4 straight y end cell cell 3 straight a plus 4 straight b end cell end table close square brackets equals open square brackets table row 1 cell negative 2 end cell 3 row 1 6 29 end table close square brackets

From the definition of equality of matrices, we have,
2    x – 3 v = 1    ...(I)
x + 4 y = 6 ...(2)
a – b= – 2 ...(3)
3a + 4 b = 29    ...(4)
(1) and (2) can be writta/i«s
2 x – 3 y – I = 0
x + 4 y – 6 = 0

therefore space space space space space space space fraction numerator straight x over denominator 18 plus 4 end fraction equals fraction numerator straight y over denominator negative 1 plus 12 end fraction equals fraction numerator 1 over denominator 8 plus 3 end fraction
therefore space space space space space space space straight x over 22 equals straight y over 11 1 over 11
therefore space space space space space space space straight x equals 22 over 11 equals 2 comma space space space straight y equals 11 over 11 equals 1

(3) and (4) can be written as
a–b+2-0
3 a + 4 b – 29 = 0

therefore space space space space space space fraction numerator straight s over denominator 29 minus 8 end fraction equals fraction numerator straight b over denominator 6 plus 29 end fraction equals fraction numerator 1 over denominator 4 plus 3 end fraction space space space space rightwards double arrow space space straight a over 21 equals straight b over 32 equals 1 over 7
therefore space space space space space space straight a equals 21 over 7 equals 3 comma space space space space space space straight b equals 35 over 7 equals 5
∴ solution is x = 2, y= 1, a = 3, b = 5.