Question

If the system of equations: $$ a_1 x + b_1 y + c_1 z = 0, \quad a_2 x + b_2 y + c_2 z = 0, \quad a_3 x + b_3 y + c_3 z = 0 $$ has only the trivial solution, then the rank of the matrix:  \[\left[ \begin{array}{ccc} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{array} \right] \] is

• A. \( 2 \)
• B. \( 1 \)
• C. \( 3 \)
• D. \( 0 \)

Hint:

For a homogeneous system to have only the trivial solution, the coefficient matrix must be of full rank, which for a 3x3 matrix means the rank is 3.

Answer 1

The Correct Option is C

To determine the rank of the matrix, we analyze the system of equations: \[ a_1 x + b_1 y + c_1 z = 0, \quad a_2 x + b_2 y + c_2 z = 0, \quad a_3 x + b_3 y + c_3 z = 0 \] The system has only the trivial solution (\( x = 0, y = 0, z = 0 \)) if and only if the determinant of the coefficient matrix is non
-zero. This implies that the matrix is full rank, meaning its rank is equal to the number of rows (or columns), which is 3. 
The matrix is: 

\[\left[ \begin{array}{ccc} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{array} \right]\]

 If the determinant of this matrix is non
-zero, the rank is 3. If the determinant were zero, the rank would be less than 3, and the system would have infinitely many solutions (non
-trivial solutions). Since the problem states that the system has only the trivial solution, the rank must be 3. Thus, the correct answer is: \[ \boxed{3} \]