Let A and B be two 2 &#x00D7;<!-- × --> 2 -matrices, and considering the equation

grenivkah3z

grenivkah3z

Answered question

2022-07-02

Let A and B be two 2 × 2-matrices, and considering the equation
A X = B
Where X is an unknown 2 × 2-matrix. What I want to do is explain why the equation is equivalent to solving two 2 × 2-systems of equations simultaneously and determine how this generalizes to n × n.
And, on the corollary, let A , B , C be three 2 × 2-matrices and consider the equation
A X + X B = C
What I want to do here explain why you can not solve this as two 2 × 2-systems of equations simultaneously

Answer & Explanation

toriannucz

toriannucz

Beginner2022-07-03Added 16 answers

A X + X B = C
( a b c d ) ( x y z w ) + ( x y z w ) ( e f g h ) = ( i j k l )
a y + b w + f x + h y = j
c y + d w + f z + h w = l
( a + e ) x + g y + b z = i
c y + f z + ( d + h ) w = l
Each of these equations involves three of the four unknowns x , y , z , w; there is no way to split the four equations into pairs so that each pair involves only two of the unknowns.
For A X = B, with all being n × n matrices, let x i be the ith column of X, 1 i n, and let b i be the ith column of B Bn A x i = b i , so there's your n systems of n equations in n unknowns.
Addison Trujillo

Addison Trujillo

Beginner2022-07-04Added 6 answers

Good answer

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