Q. 215.0( 1 Vote )

# Prove that <span

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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