Q. 215.0( 1 Vote )

Prove that <span

Answer :

To prove: is divisible by (x + y + z)


To find: the quotient



Apply C1 C1 – C2 and C2 C2 – C3




Taking common (x + y + z) from C1 and C2




Apply R1 R1 + R2 + R3





Expanding the determinant


= (x + y + z)2 (xy – z2 + yz – x2 + zx – y2){(z – y)(y – x) – (x – z)2}


= (x + y + z)2 (xy – z2 + yz – x2 + zx – y2){(zy – y2 + xy – xz) – (x2 + z2 – 2xz)}


= (x + y + z)2 (xy – z2 + yz – x2 + zx – y2)(zy – y2 + xy – xz – x2 – z2 + 2xz)


= (x + y + z)2 (zy + xy + xz – x2 – y2 – z2)2




Hence the given determinant is divisible by (x + y + z) and quotient is


OR


Given: System of equations:


8x + 4y + 3z = 19, 2x + y + z = 5, x + 2y + 2z = 7


To find: Solution of the system of the equations i.e. values of x, y and z which satisfy these equations



We know,


A = I.A where I is an identity matrix



Applying R1 R3



Applying R2 R2 – 2R1 and R3 R3 – 8R1






Applying R1 R1 – 2R2 and R3 R3 + 12R2




Applying R3 -R3 and R2 R2 – R3





To solve these equation and get values of x, y and z, we have:


AX = B where,



AX = B


X = A-1 B






Hence, solutions of the equations are x = 1, y = 2, z = 1


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