Determine if the system has a nontrivial solution (you do not need to completely solve the system). x1 – 3×2 + 7×3 = 0 -2×1 + x2 – 4×3 = 0 x1 + 2×2 + 9×3 = 0
The Correct Answer and Explanation is:
To determine if the system has a nontrivial solution, we need to analyze whether the homogeneous system of linear equations has more than just the trivial solution (i.e., all variables equal to zero). A nontrivial solution exists if the system has infinitely many solutions—this happens when the equations are linearly dependent, or equivalently, when the coefficient matrix has a determinant of zero or is not of full rank.
Given System of Equations:
x1−3×2+7×3=0−2×1+x2−4×3=0x1+2×2+9×3=0\begin{aligned} x_1 – 3x_2 + 7x_3 &= 0 \\ -2x_1 + x_2 – 4x_3 &= 0 \\ x_1 + 2x_2 + 9x_3 &= 0 \end{aligned}x1−3×2+7×3−2×1+x2−4×3x1+2×2+9×3=0=0=0
Step 1: Write the coefficient matrix:
A=[1−37−21−4129]A = \begin{bmatrix} 1 & -3 & 7 \\ -2 & 1 & -4 \\ 1 & 2 & 9 \end{bmatrix}A=1−21−3127−49
We want to determine if the homogeneous system Ax⃗=0A\vec{x} = 0Ax=0 has a nontrivial solution. This will be true if and only if the matrix AAA is not of full rank, i.e., the rows are linearly dependent.
Step 2: Perform row reduction (Gaussian elimination)
Start with:[1−37−21−4129]\begin{bmatrix} 1 & -3 & 7 \\ -2 & 1 & -4 \\ 1 & 2 & 9 \end{bmatrix}1−21−3127−49
- Add 2×Row1 to Row2:
R2=R2+2R1⇒(0,−5,10)R_2 = R_2 + 2R_1 \Rightarrow (0, -5, 10)R2=R2+2R1⇒(0,−5,10)
- Subtract Row1 from Row3:
R3=R3−R1⇒(0,5,2)R_3 = R_3 – R_1 \Rightarrow (0, 5, 2)R3=R3−R1⇒(0,5,2)
New matrix:[1−370−510052]\begin{bmatrix} 1 & -3 & 7 \\ 0 & -5 & 10 \\ 0 & 5 & 2 \end{bmatrix}100−3−557102
- Add Row2 to Row3:
R3=R3+R2⇒(0,0,12)R_3 = R_3 + R_2 \Rightarrow (0, 0, 12)R3=R3+R2⇒(0,0,12)
Final matrix:[1−370−5100012]\begin{bmatrix} 1 & -3 & 7 \\ 0 & -5 & 10 \\ 0 & 0 & 12 \end{bmatrix}100−3−5071012
This is an upper triangular matrix with 3 leading entries, so the matrix has rank 3.
Since there are 3 equations and 3 variables and the matrix has full rank, the only solution is the trivial solution.
✅ Final Answer:
No, the system does not have a nontrivial solution.
✍️ Explanation
The given system of three homogeneous linear equations involves three unknowns: x1,x2,x_1, x_2,x1,x2, and x3x_3x3. Homogeneous systems always have at least one solution—the trivial solution, where all variables are zero. The key question here is whether any nontrivial solution (a solution with at least one variable non-zero) exists.
To find this, we analyze the coefficient matrix of the system. A homogeneous system has a nontrivial solution if and only if the coefficient matrix is not of full rank, meaning its rows are linearly dependent. This would imply there are more variables than pivot positions after row reduction.
We performed Gaussian elimination on the coefficient matrix:[1−37−21−4129]\begin{bmatrix} 1 & -3 & 7 \\ -2 & 1 & -4 \\ 1 & 2 & 9 \end{bmatrix}1−21−3127−49
After row operations, we obtained a row echelon form with three pivot positions—one in each row and column. This indicates that the matrix has full rank (rank = 3). In a 3×3 system, full rank means there is exactly one solution, which for homogeneous systems, is the trivial solution.
Therefore, since the rank of the matrix equals the number of unknowns, the system is consistent and independent, and has no free variables. This implies that there is no nontrivial solution.
In conclusion, because the system’s coefficient matrix is of full rank, the system does not have a nontrivial solution.
