Question

Anyone here good with linear algebra?

Answer 1

Yes, what's your question?

Answer 2

Suppose A is an nxn matrix, whose determinant is not equal to zero and which satisfies the following condition: A^2=A. Prove that A must be equal to In, where I is the identity matrix. Cite any theorems/ definitions used.

It'd be a great help if I could get a little "push" or hint.

Answer 3

Are you taking a college course or middle school? If it's a college course, then you're further ahead than I am.

Answer 4

Sorry, high school*

Answer 5

Jzzkc, linear algebra isn't the same thing as the algebra of linear equations. :P

Answer 6

College course :(

Answer 7

Det =/= 0 <=> There Exists a nxn matrix B s.t AB = I = BA.

Answer 8

=> A^2 = AA = A
=> AAB = AB
=> AI = I
=> A = I

Answer 9

How did you get A62 = AA to become A?

Answer 10

A^2 = A. The matrix satisfies this condition.

Answer 11

and A^2 = AA.

Answer 12

Are you also famliar with span and null space?

Answer 13

Yes I suppose so.

Answer 14

How did you get from the A^2=AA=A step to the step AAB=AB?

Answer 15

I multiplied B, the matrix that has the property AB = I = AB. The Inverse of A.
You could multiply it like this => BAA = BA too if you want.