Determinant of coefficient matrix is |A| = -2 which is non-zero.
Therefore, x = y = z = 0 is the only solution.
Alternate method (Using Rank): The given system of equations can be written in the form of the single matrix equation as
We shall start reducing the coefficient matrix A to triangular form by applying only E-row transformations on it. Applying R2 → R2 - 3R1, R3 → R3 - 7R1, the given system of equations is equivalent to
Here, we find that the determinant of the matrix on the lefthand side of this equation is not equal to zero. Therefore, the rank of this matrix is 3. So, there is no need of further applying E-row transformation on the coefficient matrix. The rank of the coefficient matrix A is 3, i.e. equal to the number of unknowns. Therefore, the given system of equations does not possess any linearly independent solution. The zero solution, i.e. x = y = z = 0 is the only solution of the given system of equations.
