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Consider the system of equations a(1) ...

Consider the system of equations
`a_(1) x + b_(1) y + c_(1) z = 0`
`a_(2) x + b_(2) y + c_(2) z = 0`
`a_(3) x + b_(3) y + c_(3) z = 0`
If `|(a_(1),b_(1),c_(1)),(a_(2),b_(2),c_(2)),(a_(3),b_(3),c_(3))| =0`, then the system has

A

more than one solution

B

one trivial and one non trivial solution

C

no solution

D

only trivial solution (0,0,0)

Text Solution

AI Generated Solution

The correct Answer is:
To solve the given system of equations, we need to analyze the conditions under which the system has solutions. The equations are: 1. \( a_1 x + b_1 y + c_1 z = 0 \) 2. \( a_2 x + b_2 y + c_2 z = 0 \) 3. \( a_3 x + b_3 y + c_3 z = 0 \) We are also given that the determinant of the coefficients of these equations is zero: \[ D = \begin{vmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{vmatrix} = 0 \] ### Step 1: Understanding the Implication of the Determinant Being Zero When the determinant \( D \) of the coefficients is zero, it indicates that the rows (or columns) of the matrix are linearly dependent. This means that at least one of the equations can be expressed as a linear combination of the others. **Hint:** Remember that a zero determinant implies linear dependence among the equations. ### Step 2: Analyzing the System of Homogeneous Equations Since all the equations are homogeneous (equal to zero), we can conclude that the trivial solution \( (x, y, z) = (0, 0, 0) \) is always a solution. However, because the determinant is zero, there must also be non-trivial solutions. **Hint:** Homogeneous equations always have the trivial solution, but linear dependence can lead to non-trivial solutions. ### Step 3: Conclusion About the Number of Solutions Given that the equations are linearly dependent and homogeneous, the system will have infinitely many solutions. This is because the presence of linear dependence means that there are free variables that can take on multiple values, leading to an infinite number of solutions. **Hint:** Linear dependence in homogeneous equations typically results in infinitely many solutions. ### Final Answer Thus, the system of equations has more than one solution, specifically infinitely many solutions. **Correct Option:** More than one solution (infinitely many solutions). ---
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