Page 399 - 35Linear Algebra

G.7 Determinants 399

 
1
 . 
.
 
.
 
1 λ
 
 
i
S (λ) =  . . 
j


 . 
1
 
 
 
.
.
 
.
 
1
So to calculate their determinants, we just have to apply the above list
of what happens to the determinant of a matrix under row operations to the
determinant of the identity. This yields
i
i
i
det E = −1 , det R (λ) = λ , det S (λ) = 1 .
j
j
Determinants and Inverses
Lets figure out the relationship between determinants and invertibility. If
we have a system of equations Mx = b and we have the inverse M −1 then if we
multiply on both sides we get x = M −1 Mx = M −1 b. If the inverse exists we
can solve for x and get a solution that looks like a point.
So what could go wrong when we want solve a system of equations and get a
solution that looks like a point? Something would go wrong if we didn’t have
enough equations for example if we were just given
x + y = 1
or maybe, to make this a square matrix M we could write this as
x + y = 1
0 = 0

1 1
The matrix for this would be M = and det(M) = 0. When we compute the
0 0
determinant, this row of all zeros gets multiplied in every term. If instead
we were given redundant equations

x + y = 1
2x + 2y = 2

1 1
The matrix for this would be M = and det(M) = 0. But we know that
2 2
with an elementary row operation, we could replace the second row with a row

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