+0  
 
0
221
1
avatar

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
 #1
avatar
0

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

21 Online Users

New Privacy Policy

We use cookies to personalise content and advertisements and to analyse access to our website. Furthermore, our partners for online advertising receive information about your use of our website.
For more information: our cookie policy and privacy policy.