The system of equations

x + 2y + z = 0

x – z = 0

x + y = 0

has

Concept:

Consider the system of m linear equations

a11 x1 + a12 x2 + … + a1n xn = 0

a21 x1 + a22 x2 + … + a2n xn = 0

…

am1 x1 + am2 x2 + … + amn xn = 0

The above equations contain the n unknowns x1, x2, …, xn. To determine whether the above system of equations is consistent or not, we need to find the rank of the following matrix.

A is the coefficient matrix of the given system of equations.

We can find the consistency of the given system of equations as follows:

(i) If the rank of matrix A is equal to the number of unknowns, then the system has only a trivial zero solution.

The rank of A = n

(ii) If the rank of matrix A is less than the number of unknowns, then the system has an infinite number of solutions.

The rank of A < n

Calculation:

Given:

The given system of equations can be represented in a matrix form as shown below.

C1 → C1 + C3,

C1 = C2, Therefore |A|3× 3 = 0, Rank ≠ 3

Determinant of the matrix order 2 ≠ 0, Therefore rank of matrix A = 2

Rank of matrix A = 2 < n = 3

As rank of matrix is less than the variables (n). Hence the given system of linear equations has infinitely many solutions.