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# I promise, last matrix problem.

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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.

Jul 21, 2017

### 1+0 Answers

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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)

Jul 21, 2017