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78 Chapter Three: Determinants

n
Prove that Z^n-i = 0 and D 2n — (—ab) .
j
9. Let D n be the nxn determinant whose (i,j) entry is i + .
Show that D n = 0 if n > 2. [Hint: use row operations].
10. Let u n denote the number of additions, subtractions and
multiplications needed in general to evaluate an n x n deter-
minant by row expansion. Prove that u n = nu n _i + 2n — 1.
Use this formula to calculate u n for n = 2,3,4.

3.3 Determinants and Inverses of Matrices
An important property of the determinant of a square
matrix is that it tells us whether the matrix is invertible.

Theorem 3.3.1
An nxn matrix A is invertible if and only if det( A) ^ 0.

Proof
By 2.3.2 there are elementary matrices E\,E2, • • • ,Ek such
that the matrix R = E^Ek-i • • • E^E\A is in reduced row
echelon form. Now observe that if E is any elementary nxn
matrix, then det(EA) = cdet(^4) for some non-zero scalar c;
this is because left multiplication by E performs an elementary
row operation on A and we know from 3.2.3 that such an
operation will, at worst, multiply the value of the determinant
by a non-zero scalar. Applying this fact repeatedly, we obtain
det(-R) = det(Ek • • • E2E1A) = ddet(A) for some non-zero
scalar d. Consequently det(-A) 7^ 0 if and only if det(i?) ^ 0.
Now we saw in 2.3.5 that A is invertible precisely when
R = I n . But, remembering the form of the matrix R, we
recognise that the only way that det(-R) can be non-zero is if
R = I n. Hence the result follows.

Example 3.3.1
The Vandermonde matrix of Example 3.2.4 is invertible if and
only if all different.