Given that the A_11 and A_22 in RR^(3 xx 3) are invertible, A_21 in RR^(3 xx 3), and b_1,b_2,x_1,x_2 in RR^3, then solve for x_1 and x_2 from [[A_11,0],[A_21,A_22]] [[x_1],[x_2]]=[[b_1],[b_2]] What are x_1 and x_2 in terms of A_11,A_21,A_22,b_1,b_2?

gorgeousgen9487 2022-07-13 Answered
Given that the A 11 and A 22 R 3 x 3 are invertible, A 21 R 3 x 3 , and b 1 , b 2 , x 1 , x 2 R 3 , then solve for x 1 and x 2 from
[ A 11 0 A 21 A 22 ] [ x 1 x 2 ] = [ b 1 b 2 ]
What are x 1 and x 2 in terms of A 11 , A 21 , A 22 , b 1 , b 2 ?
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Answers (1)

tilsjaskak6
Answered 2022-07-14 Author has 14 answers
Split it up into two equations
A 11 x 1 = b 1 A 21 x 1 + A 22 x 2 = b 2
Solve for x 1 and in the first equation and use it in the second
x 1 = A 11 1 b 1 A 22 x 2 = b 2 A 21 A 11 1 b 1
Solve for x 2
x 2 = A 22 1 ( b 2 A 21 A 11 1 b 1 )
Re-combine the solution
[ x 1 x 2 ] = [ A 11 1 0 A 22 1 A 21 A 11 1 A 22 1 ] [ b 1 b 2 ]

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