Question

A system of homogeneous equations for n unknowns has a trivial solution if the rank of A is:​

Answer 1

Answer:

this is the correct answer

Answer 2

For a system of homogenous equations in 'n' unknowns to have a trivial solution, the rank of A must be equal to 'n' the number of unknowns.

Explanation:

• Let 'A' be a matrix representing a system of homogenous equations in 'n' variables such that,  
•                      $AX=0\ \ \ \ \ \ \ ->X=\left[\begin{array}{ccc}x_1\\x_2\\.\\.\\x_n\end{array}\right]$    
• Then this system of equations may have a trivial solution such that $X=0\ \ \ ->x_1=x_2=...=x_n=0$  
• The number of linearly independent rows or columns in a given matrix is called the rank of that matrix.
• It is denoted by the 'rho' symbol, $\rho(A)$.
• Hence, in order to obtain a trivial solution for the given homogenous system of the equation the rank of matrix A should be equal to the number of unknowns -> $\rho (A)= n$