Determine whether or not the determinant is 0.

Question

Given a 3 × 3 matrix B with real entries, suppose that the system of linear equations determined by  B(x1 x2 x3) = (0 1 1) (they should be column matrices.) has infinitely many solutions. Determine whether or not det(B) = 0, carefully
justifying your answer.

 

Answer 1

Let X be the column matrix (x₁  x₂  x₃)  and let C be the column matrix  

(0   1   1).  I typed them as rows, but they should be written vertically as single columns. 

 

Then BX = C

 

Suppose that det(B) ≠ 0.  Then B^-1 exists.

 

So, since BX = C, we have B^-1(BX) = B^-1C

 

                            Therefore,  X = B^-1C

 

So, the system of equations has a UNIQUE solution.  

 

This contradicts the given information that the system has infinitely many solutions.

 

So, our assumption that det(B) ≠ 0 must have been incorrect.  

 

Thus, det(B) = 0