# (Please help ) If A² = I, prove det A = ±1

1. May 23, 2008

### chapone

(Please help!!) If A² = I, prove det A = ±1

If A² = I, show that det A = ±1

This book is very unclear, but I am assuming by "I" they mean the identity matrix with a size of 2x2. I have tried putting in A for row 1 column 1 - B for 1,2 - C for 2,1 and D for 2,2 multiplying and setting the results equal to the values of the identity matrix. I thought I was close, but now am doubting that I am going about this the right way. Any help is MUCH appreciated! Thank you!

2. May 23, 2008

### Hurkyl

Staff Emeritus
Well, in this case, doing matrix arithmetic is going to be a lot easier than doing scalar arithmetic on the entries of the matrix.

3. May 23, 2008

### tiny-tim

Welcome to PF!

Hi chapone! Welcome to PF!

What makes you think they mean 2x2?

This theorem is true for any n x n matrix.

Do you know any formulas for determinants (for example, for det (AB))?

4. May 26, 2008

### asdfggfdsa

Take the determinant of both sides of A$$^{2}$$ = I.

(Any don't make any assumptions beyond what the book gives you.)

5. Oct 14, 2010

### john the gree

Re: (Please help!!) If A² = I, prove det A = ±1

a² = 1
a² - 1 = 0
(a - 1) (a + 1) = 0
a=1 and a = -1
then a = ±1

6. Oct 14, 2010

### Staff: Mentor

Re: (Please help!!) If A² = I, prove det A = ±1

This is all well and good for a real number a, but the OP is working with a matrix A, not a scalar. As such, A $\neq$ 1.

7. Oct 14, 2010

### Fredrik

Staff Emeritus
Re: (Please help!!) If A² = I, prove det A = ±1

chapone got 3 good answers already, so I'll just add to what Mark44 said (also a good post, but not an answer for chapone) by providing a counterexample that shows that John's argument gets the wrong result for matrices:

$$\begin{pmatrix}0 & 1\\ 1 & 0\end{pmatrix}\begin{pmatrix}0 & 1\\ 1 & 0\end{pmatrix}=\begin{pmatrix}1 & 0\\ 0 & 1\end{pmatrix}$$

8. Oct 23, 2010

### vigvig

Re: (Please help!!) If A² = I, prove det A = ±1

If A^2=1, then A is a matrix of order 2, which means that A is invertible.

Last edited by a moderator: Oct 23, 2010
9. Oct 23, 2010

### D H

Staff Emeritus
Re: (Please help!!) If A² = I, prove det A = ±1

There are multiple meanings of the term "order", and your meaning here is not the meaning usually meant for matrices.

Last edited: Oct 23, 2010
10. Oct 24, 2010

### vigvig

Re: (Please help!!) If A² = I, prove det A = ±1

What do you mean? Order in the sense of "group order", meaning A generates a group of order 2. Implying A must be invertible

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