# Solving simultaneous equations

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

Mark44
Mentor
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.