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how does one use the determinant of the coefficient matrix of a system to determine if the system has nontrivial solutions or not?
No, this is incorrect. Also, an equation is never equal to anything. For example, x + 5 = 2 is an equation, but what is it equal to? There is always an = in an equation, but that indicates that two expressions have the same value.determinant = 0, homogeneous equation equals zero... therefore trivial solution
determinant not to equal 0, homogeneous equation don't equal 0.... therefore nontrivial solution?
No. In that case the matrix of coefficients is not square (has more columns than rows). The determinant is defined only for square matrices.what about a homogeneous system of equations with more unknowns than equations, does the above also apply?