Quandaries and Queries


My question is how to solve this matrice question?
: Let

      | 1 -1 0 |
A= | 2 -1 2 |
      | a  b  c|

where a, b, c are constant real numbers. For what values of a, b, c is A invertible? [Hint: Your answer should be an equation in a, b, c which satisfied if and only if A is invertible.]

Thank you




There are two good approaches to solving this problem:

One is to row reduce the matrix (you can multiple row 1 by a and subtract it from row 3 to get a zero in the 3,1 position, etc). For this matrix, you can get a leading 1 in the 1,1 and 2,2 positions without involving any variables. However if you do things carefully, you will end up with the 3,1 and 3,2 entries being zero and an equation in a,b,c in the 3,3 position. The matrix is invertible if and only if this entry (equation) is nonzero.

A second method, if you know how to take the determinant of a matrix, would be to take the determinant of this matrix. This will give you an equation in a,b, and c, and again, the matrix is invertible if and only if this the determinant is nonzero.

I hope one of these methods helps you to solve the problem.



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