+0

# I promise, last matrix problem.

0
51
1

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
Sort:

#1
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

### 8 Online Users

We use cookies to personalise content and ads, to provide social media features and to analyse our traffic. We also share information about your use of our site with our social media, advertising and analytics partners.  See details