(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.
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.