SOLUTION: Why is the matrix {{{(matrix(2,2,x-1,-1,-3,x-5))}}} not invertible for all real numbers x?

Algebra ->  Matrices-and-determiminant -> SOLUTION: Why is the matrix {{{(matrix(2,2,x-1,-1,-3,x-5))}}} not invertible for all real numbers x?       Log On


   

Question 1032073: Why is the matrix %28matrix%282%2C2%2Cx-1%2C-1%2C-3%2Cx-5%29%29
not invertible for all real numbers x?
Answer by Edwin McCravy(20055) About Me  (Show Source):
You can put this solution on YOUR website!
Why is the matrix %28matrix%282%2C2%2Cx-1%2C-1%2C-3%2Cx-5%29%29
not invertible for all real numbers x?
A square matrix whose determinant is 0 is not invertible

abs%28matrix%282%2C2%2Cx-1%2C-1%2C-3%2Cx-5%29%29%22%22=%22%22%28x-1%29%28x-5%29-%28-1%29%28-3%29%22%22=%22%22x%5E2-6x%2B5-3%22%22=%22%22x%5E2-6x%2B2

If x%5E2-6x%2B2%22%22=%22%220 then the matrix is not invertible.

x%22%22=%22%22%28-%28-6%29+%2B-+sqrt%28%28-6%29%5E2-4%281%29%282%29+%29%29%2F%282%281%29%29+

x%22%22=%22%22%286+%2B-+sqrt%2836-8%29%29%2F2+

x%22%22=%22%22%286+%2B-+sqrt%2828%29%29%2F2+

x%22%22=%22%22%286+%2B-+sqrt%284%2A7%29%29%2F2+

x%22%22=%22%22%286+%2B-+2sqrt%287%29%29%2F2+

x%22%22=%22%226%2F2+%2B-+2sqrt%287%29%2F2+

x%22%22=%22%223+%2B-+sqrt%287%29+

The matrix is not invertible if x%22%22=%22%223+%2B+sqrt%287%29+ or x%22%22=%22%223+-+sqrt%287%29+.

Edwin