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Prove that for any 2 x 2 matrix A, Acan be written in the linear form aA + bI 

 

I = (identity matrix).

 

I, think I,  know this is true, I just can't imagine how to prove it.

Guest Jul 21, 2017
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you should try to devlop this

 \(\begin{pmatrix} a&b \\ c&d \end{pmatrix} ^{2} =e\begin{pmatrix} a&b \\ c&d \end{pmatrix} +f\begin{pmatrix} 1&0 \\ 0&1 \end{pmatrix}\)

It should give the following equations:

\(\left\{\begin{matrix} a^2+bc=ae+f \\ ab+bd=eb \\ ac+dc=ec \\ bc+d^2=f+ed \end{matrix}\right.\)

By solving the second equation, you can easily find \(e=a+d\)

wich also work with the third equation. 

with the first and the fourth we can find \(f=bc-ad\)

 

Therefor we have shown that this system as always a solution for e=a+d and f=bc-ad=-det(A)

Guest Jul 21, 2017

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