如果考慮 A x = 0
那麼如果 A 是 nonsingular
就表示 A 是 invertible
也就是 det(A) 不等於零
或者 rank(A) = n
在這種情況下, x = 0 是唯一的解,
也就是說, 根本就沒有其它的解存在。
這樣講是沒錯，但是事實上證明「A可逆 <=> Ax = 0只有唯一解x=0」可以直接證明。
nxn
Theorem. Let A in F . Then the following are equivalent.
(1) A is invertible. (2) Ax = 0 implies x = 0.
(3) A is row equivalent to I. (4) A can be expressed as a product of
elementary matrices.
proof: (1)=>(2) Clearly, x= 0 is a solution of Ax = 0. For uniqueness,
if y is another solution of Ax = 0,
-1
then apply A to Ax = Ay, we have x = y.
(2)=>(3) If A is not row equivalent to I. The reduced row echeloned
A' of A must have at least one zero row. This implies that
Ax = 0 has infinitely many solutions, contradicting the fact
that Ax = 0 has only trivial solution x = 0.
Hence, A is row equivalent to I.
(3)=>(4) Since A is row equivalent to I and each row operation on A
can be replaced by a elementary matrix, let E be the finite
product of such elementary matrices. Then EA = I.
-1
That is, A = E which is a product of elementary matrices.
(4)=>(1) By (4), our result follows form the fact that each elmentary
matrix is invertible.
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