Find a 2 X 2 matrix B such that $open square brackets table row 2 5 row cell negative 3 end cell 7 end table close square brackets space space straight B equals open square brackets table row 17 cell negative 1 end cell row 47 cell negative 13 end cell end table close square brackets$

Solution

Let space straight B equals space open square brackets table row straight x straight y row straight z straight t end table close square brackets therefore space space space open square brackets table row 2 5 row cell negative 3 end cell 7 end table close square brackets space straight B equals space minus open square brackets table row 2 5 row cell negative 3 end cell 7 end table close square brackets space space open square brackets table row straight X straight y row straight Z straight t end table close square brackets equals open square brackets table row 17 cell negative 1 end cell row 47 cell negative 13 end cell end table close square brackets rightwards double arrow open square brackets table row cell 2 straight x plus 5 straight z end cell cell 2 straight y plus 5 straight t end cell row cell negative 3 straight x plus 7 straight z end cell cell negative 3 straight y plus 7 straight t end cell end table close square brackets equals space open square brackets table row 17 cell negative 1 end cell row 47 13 end table close square brackets

By definition of equality of matrices, we get
2x + 5 z = 17    ...(1)
–3 x + 7 z =47    ...(2)
2y + 5 t =–1    ...(3)
–3y + 7 t = – 13    ...(4)
Multiplying (1) by 3 and (2) by 2, we get,
6 x + 15z = 51    ...(5)
–6x + 14 z = 94    ...(6)
Adding (5) and (6), we get,
29 z = 145 or z = 5
∴ from (1 )2 x + 25 = 17 or 2.x = – 8 ⇒ x = – 4
Multiplying (3) by 3 and (4) by 2, we get,
6.y + 15t =.– 3    ...(7)
– 6 y+ 14 t = – 26    ...(8)
Adding (7) and (8), we get,
29 t = – 29 or t = –1
∴ from (3) 2y – 5 = – 1 or 2 y = 4 ⇒ y = 2
we have
x = – 4, y = 2, z = 5, t = – 1

therefore space straight B equals space open square brackets table row cell negative 4 end cell 2 row 5 cell negative 1 end cell end table close square brackets

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Matrices

Find a 2 X 2 matrix B such that $open square brackets table row 2 5 row cell negative 3 end cell 7 end table close square brackets space space straight B equals open square brackets table row 17 cell negative 1 end cell row 47 cell negative 13 end cell end table close square brackets$

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