Determinant of Matrix | Generated by AI
If the determinant of matrix \( A \) is zero, the system may have infinitely many solutions because it indicates that the matrix is singular (non-invertible), meaning its rows or columns are linearly dependent. Let’s explore why this leads to infinite solutions.
1. Determinant and Rank
- The determinant of a square matrix \( A \) tells us whether the matrix is invertible.
- If \( \det(A) \neq 0 \), the matrix is invertible, and the system has a unique solution.
- If \( \det(A) = 0 \), the matrix is singular (non-invertible), meaning the system has either no solution or infinitely many solutions.
- The rank of a matrix is the number of linearly independent rows or columns.
- If the number of nonzero rows (rank) is less than the number of variables, we get free variables, leading to infinitely many solutions.
2. Why Infinite Solutions?
Case: Homogeneous System \( A\mathbf{x} = 0 \)
- A homogeneous system always has the trivial solution \( \mathbf{x} = 0 \).
- If \( \det(A) = 0 \), it means that at least one row is dependent on another.
- This means at least one free variable exists, allowing an infinite number of solutions.
Example:
[
\begin{aligned}
x + y + z &= 0
2x + 2y + 2z &= 0
\end{aligned}
]
- The second equation is just twice the first one → only one independent equation.
- We have three variables but one independent equation, so we have free variables → infinitely many solutions.
Case: Non-Homogeneous System \( A\mathbf{x} = \mathbf{b} \)
- If \( \det(A) = 0 \), the system is either inconsistent (no solution) or has infinitely many solutions.
- The system has infinitely many solutions when the number of independent equations is less than the number of variables.
Example:
[
\begin{aligned}
x + y + z &= 5
2x + 2y + 2z &= 10
\end{aligned}
]
- The second equation is just twice the first one.
- One independent equation with three variables → infinitely many solutions (parametric solution form).
Conclusion
- If \( \det(A) = 0 \), at least one row is dependent, meaning we have free variables.
- Free variables allow for an infinite number of solutions in both homogeneous and some non-homogeneous cases.
- If \( A \) is singular and the system is inconsistent (e.g., conflicting equations like \( x + y = 2 \) and \( x + y = 3 \)), there is no solution instead of infinite solutions.
Let me know if you need further clarification! 🚀