Solving simultaneous equations

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In solving a system of simultaneous equations represented in matrix form as Ax = 0, the determinant of the matrix A plays a crucial role. For a unique solution to exist, the determinant must be nonzero, indicating that A has an inverse. If the determinant vanishes (det(A) = 0), it signifies that A lacks an inverse, resulting in an infinite number of solutions. This distinction is essential for understanding the nature of solutions in linear algebra. Therefore, the condition of the determinant directly influences the solvability of the system.
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when solving a system of simultaneous equations in matrix from (with the LHS = 0) why does the determinant of the matrix need to vanish?

thanks
 
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lavster said:
when solving a system of simultaneous equations in matrix from (with the LHS = 0) why does the determinant of the matrix need to vanish?

thanks

I'm guessing you're talking about a matrix equation Ax = 0, where A is an n x n matrix containing the coefficients of the variables, x is a column vector with n entries, and 0 is a column vector whose n entries are all zero. (BTW, you almost always see the zero vector on the right side of the equation, not the left.)

For a unique solution to the system, the determinant of A must be nonzero; i.e., must NOT vanish. If det(A) is not zero, then A has an inverse, so the solution to the system is obtained by multiplying both sides of the equation by A-1.

Ax = 0 ==> A-1Ax = A-10 == > x = 0

If the determinant of A vanishes (i.e., det(A) = 0), then A does not have an inverse, which means in this case that there are an infinite number of solutions.
 

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