Solve:

(i) x+y-2z =0 (ii)2x+3y+4z =0 (iii)3x+y+z =0 (iv) x+2y-3z = -4

2x+y-3z =0 x+y+z =0 x-4y+3z =02x+3y+2z =2

5x+4y-9z =0 2x-y+3z =0 2x+5y-2z =0 3x-3y-4z =11

(1)  Given homogeneous equations are –

Let  AX = 0  be a homogeneous system of  n  linear equations with  n  unknowns.

Where

Applying  c₂ → c₂ – c₁  and  c₃ → c₃ + 2 c₁ ,  we get

=  1 (–1 + 1) = 0

i.e.  Matrix A  is singular.

Then,  the system has infinitely many solutions.

Put  z  =  k  (any real number)  and solve first two equations for  x  and  y.

          x + y = 2k  .....  (4)

and  2x + y = 3k  .....  (5)

or  AX  =  B

So,  A^–1  will exist.

Now put  x = y = z = k  in  (3),  we get –

5k + 4k – 9k = 0

⇒  0 = 0

which is true.

Hence,  x = k ,  y = k ,  z = k  where  k  is any real number satisfy the given system of equations

Q4  Given system of equations is –

The given system can be written  as –

AX = B

Since    is non-singular and hence  a unique solution given by 

X = A^–1 B  will exist.

∴  X = A^–1 B

⇒  x = 3,  y = –2,  z = 1

(2)  and  (3)  queries can  be solved in the same way.  If you face any problem,  please do get back to us.