Determinant of Matrix | Generated by AI

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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


2. Why Infinite Solutions?

Case: Homogeneous System \( A\mathbf{x} = 0 \)

Example:
[ \begin{aligned} x + y + z &= 0
2x + 2y + 2z &= 0 \end{aligned} ]


Case: Non-Homogeneous System \( A\mathbf{x} = \mathbf{b} \)

Example:
[ \begin{aligned} x + y + z &= 5
2x + 2y + 2z &= 10 \end{aligned} ]


Conclusion

Let me know if you need further clarification! 🚀


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