# Matrix such that $B^2 + B - I = 0$

The following is an old exam question I'm trying solve but I just can't. Any help would be much appreciated. The teachers has provided answers to all questions on the exam besides this one..

A quadratic matrix $B$ satisfies the equation $B^2 + B - I = 0$ where $I$ is an identity matrix with dimensions $N\times N$ and where $0$ is the null matrix with the dimensions $N\times N$.

a) What size is the matrix $B$?
b) Show that $I + B$ is the inverse matrix of $B$.
c) Show that $B^3 = -I + 2B$.

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Note that the 1x1 matrix with the value (sqrt(5)-1)/2 = 0.6180... is a valid example for B. – Sjoerd Nov 19 '11 at 19:57

a) The size of the matrix $B$ must be the same as $I$ so it is $N \times N$.

b) $(I+B) B=IB +B^2=B+B^2=I$ using the equation and so $I+B$ is the inverse of $B$.

c) $B^3=B B^2=B(I-B)= B -B^2=B-(I-B)=2B-I$ using the equation $B^2+B-I=0$.

If you need more details just tell me.

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I appreciate that you showed the steps in detail. Thanks! – grandnasty Nov 19 '11 at 19:13

Note that you can only add matrices if the are of the same size. In this case we have a quadratic matrix $B$, so it is of the format $n\times n$ and we want to find out what $n$ is. We have that $B^2+B-I=0$ and we can only add $B$ to $-I$ if $n=N$. Also, $B^2$ has size $N\times N$.

In order to verify (b) calculate $(I+B)\cdot B=IB+B^2=B+B^2$. Now use the euqation $B^2+B-I$.

For (c) multiply the equation $B^2+B-I=0$ by $B$.

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Thanks Oliver, much appreciated! – grandnasty Nov 19 '11 at 19:14