Question

一道线性代数题

一道线性代数题求指教

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Answer 1

|A*| = 8 = |A|^(4-1) = |A|^3 , |A| = 2.
原式左乘以 A， 得 AB = B + 3A， (A-E)B = 3A， B = (A-E)^(-1)(3A).
AA* = |A|E, A = |A|(A*)^(-1).
(A*, E) =
[1 0 0 0 1 0 0 0]
[0 1 0 0 0 1 0 0]
[1 0 1 0 0 0 1 0]
[0 -3 0 8 0 0 0 1]
初等行变换为
[1 0 0 0 1 0 0 0]
[0 1 0 0 0 1 0 0]
[0 0 1 0 -1 0 1 0]
[0 0 0 8 0 3 0 1]
初等行变换为
[1 0 0 0 1 0 0 0]
[0 1 0 0 0 1 0 0]
[0 0 1 0 -1 0 1 0]
[0 0 0 1 0 3/8 0 1/8]
A = |A|(A*)^(-1) = 2(A*)^(-1)
[ 2 0 0 0]
[ 0 2 0 0]
[-2 0 2 0]
[0 3/4 0 1/4]
[(A-E), (3A)] =
[ 1 0 0 0 6 0 0 0]
[ 0 1 0 0 0 6 0 0]
[-2 0 1 0 -6 0 6 0]
[0 3/4 0 -3/4 0 9/4 0 3/4]
初等行变换为
[ 1 0 0 0 6 0 0 0]
[ 0 1 0 0 0 6 0 0]
[ 0 0 1 0 6 0 6 0]
[0 0 0 -3/4 0 -9/4 0 3/4]
初等行变换为
[ 1 0 0 0 6 0 0 0]
[ 0 1 0 0 0 6 0 0]
[ 0 0 1 0 6 0 6 0]
[0 0 0 1 0 3 0 -1]
B =
[6 0 0 0]
[0 6 0 0]
[6 0 6 0]
[0 3 0 -1]

追问

你好 为什么|A|=2？

追答

|A*| = 8
|A*| = |A|^(4-1) = |A|^3 = 8,
则 |A| = 2.