write B as a linear combination of the other matrices, if possible. B=[[2,-2,3],[0,0,-2],[0,0,2]] A_1=[[1,0,0],[0,1,0],[0,0,1]] A_2=[[0,1,1],[0,0,1],[0,0,0]] A_3=[[-1,0,-1],[0,1,0],[0,0,-1]] A_4=[[1,-1,1],[0,-1,-1],[0,0,1]]

Carol Gates 2021-01-13 Answered
write B as a linear combination of the other matrices, if possible.
B=[[2,2,3],[0,0,2],[0,0,2]]
A1=[[1,0,0],[0,1,0],[0,0,1]]
A2=[[0,1,1],[0,0,1],[0,0,0]]
A3=[[1,0,1],[0,1,0],[0,0,1]]
A4=[[1,1,1],[0,1,1],[0,0,1]]
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Expert Answer

hosentak
Answered 2021-01-14 Author has 100 answers
Matrix solution:

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Jeffrey Jordon
Answered 2021-09-30 Author has 2495 answers

Consider the linear combination of matrices as:

B=αA1+BA2+γA3+δA4, where α,β,γ,δ are constants.

Substitute the matrices of B,A1,A2,A3,A4 to find constants:

[223002002]=α[100010001]+β[001001000]+γ[100010001]+δ[111011001]

[223002002]=[αγ+δβδβγ+δ0α+γδβδ00αγ+δ]

By equation component equations:

αγ+δ=2 (1)

βδ=2 (2)

βγ+δ=3 (3)
α+γδ=0 (4)

Adding (1) and (4) 2α=2α=1

From equation (2) β=δ2

Therefore, equation (3) becomes

δ2γ+δ=3

2δ=5+γ

δ=5+γ2

Substitute above value in equation (4), if becomes,

1+γ5+γ2=0

2+2γ5γ=0

γ=3

Therefore, δ=5+γ2=5+32=4

Frome equation (2),

β=42=2

Substitute the values of constants

Therefore, the linear combination of matrix B is given by

B=1A1+2A2+3A3+4A4

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