It is known that a = (α, β, γ), B = (α + β + γ, α + 2 β + 4 γ, α + 3 β + 9 γ), where α, β and γ are 3-dimensional sequence vectors, | a | = m, find | B|

It is known that a = (α, β, γ), B = (α + β + γ, α + 2 β + 4 γ, α + 3 β + 9 γ), where α, β and γ are 3-dimensional sequence vectors, | a | = m, find | B|

|B | = | α + β + γ, α + 2 β + 4 γ, α + 3 β + 9 γ | (the first column is multiplied by - 1 and added to the second and third column) = | α + β + γ, β + 3 γ, 2 β + 8 γ, | (the second column is multiplied by - 2 and added to the third column) = | α + β + γ, β + 3 γ, 2 γ | (after the second column is extracted from the third column, the third column is multiplied by - 1 and added to the first column, and the third column is multiplied by - 3 and added to the second column)

Let 4-order matrix A = [α, γ 2, γ 3, γ 4], B = [β, γ 2, γ 3, γ 4], where α, β, γ 2, γ 3, γ 4 are all 4-dimensional column vectors, and the determinant | a | = 4, | B | = 1 is known, then the determinant | a + B ||=______ ．

A + B = [α + β, 2Y2, 2y3, 2y4] = 8 [α + β, Y2, Y3, Y4] & nbsp; & nbsp; so: | a + B | = 8 | α + β, Y2, Y3, Y4 | = 8 (| a | + | B |) = 40

A is a 3-order square matrix, α is a 3-dimensional column vector, and α, a α, a & # 178; α is linearly independent. Know a & # 179; α = a α. Find the determinant of (a + 2e)

Let B = PAP ^ (- 1),
P=(α,Aα,A²α)
Then BP = PA = (a α, a ^ 2 * α, a ^ 3 * α)
=(Aα,A^2*α,Aα)
In the above formula, a α and a ^ 2 * α are linearly independent
Then it can be obtained by the multiplication of the matrix
B=(0 1 0,0 0 1,0 1 0)
And because B = PAP ^ (- 1)
So a = P ^ (- 1) BP
A+2E=P^(-1)BP+2E
=P^(-1)BP+2P^(-1)P
=P^(-1)（B+2E)P
So B + 2E is the similarity matrix of a + 2E
And because similar matrices have the same determinant
So the determinant of (a + 2e) can be changed into the determinant of (B + 2e)
complete