(0 )2 1436587@9BAC9EDGF4HI 7P5QHRFTSVU0DXW6F4H
(0 )2 1436587@9BAC9EDGF4HI 7P5QHRFTSVU0DXW6F4H
(0 )2 1436587@9BAC9EDGF4HI 7P5QHRFTSVU0DXW6F4H
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(det A) (det B) = 0 ⇐⇒ (det A) = 0 (det B) = 0<br />
A ⇐⇒ ¢ ¨ <br />
B<br />
¡ ¨ ¥ A B n ¡ <br />
<br />
¥¤ P n ¨ B = P −1 AP<br />
§ <br />
<br />
¨¥ ¨ A B n ¡ <br />
<br />
¡<br />
det B = det (P −1AP ) = det (P −1 ) det A det P = 1<br />
det P<br />
= det A<br />
P n ¨ B = P −1 AP<br />
<br />
T AA = £¢ <br />
In <br />
¥¤ <br />
p p ∈ IN 0£¢ A = det A<br />
<br />
= 0<br />
¡ ¨ A n<br />
det A ∈ {−1, 1}<br />
¤ £¢ <br />
det A det P<br />
¨¥ T<br />
¨<br />
AA = £¢<br />
In<br />
det AAT = det (In) ⇐⇒ (det A) det(AT ) = det (In)<br />
⇐⇒ (det A) (det A) = 1<br />
⇐⇒ (det A) 2 − 1 = 0<br />
⇐⇒ (det A − 1) (det A + 1) = 0<br />
⇐⇒ det A = 1 det A = −1<br />
¥¤ 0£¢ p p ∈ IN A =<br />
det (Ap ¡¡ ¡<br />
<br />
) = det 0 ⇐⇒ det AA · · · A<br />
p<br />
= 0<br />
⇐⇒ (det A) (det A) · · · (det A) = 0<br />
<br />
¡ ¡¡ p<br />
⇐⇒ (det A) p = 0 ⇐⇒ det A = 0<br />
65