Question

If A is invertible, then so is A^3 +I
True of false? why?

Answer 1

For a matrix, A^3 = A*A*A. Since A^3 is invertible, then A*A*A*B = I for some matrix B.

Matrix multiplication is associative even though it isn't commutative, therefore A*A*A*B = A*A*(A*B) = I so define the matrix C = A*B.

Now, A*A*C = A*A*A*B = A^3*B = I so A^2 is also invertible; its inverse is C.

Answer 2

what happens if you let

\[A=\left[\begin{matrix}-1 & 0 \\ 0 & -1\end{matrix}\right]\]

Answer 3

the A I gave above is invertible

is \(A^3+I\) invertible?

Answer 4

oh sorry i did find the inverse but i am still kind of lost...the person above gave me an explanation i'm trying to understand it

Answer 5

if \(A=\left[\begin{matrix}-1 & 0 \\ 0 & -1\end{matrix}\right]\)

then \(A^3+I\)

\[=\left[\begin{matrix}-1 & 0 \\ 0 & -1\end{matrix}\right]^3+\left[\begin{matrix}1 & 0 \\ 0 & 1\end{matrix}\right]=\left[\begin{matrix}-1 & 0 \\ 0 & -1\end{matrix}\right]+\left[\begin{matrix}1 & 0 \\ 0 & 1\end{matrix}\right]=\left[\begin{matrix}0 & 0 \\ 0 & 0\end{matrix}\right]\]

Answer 6

im cnfused this also makes sense. im trying to go over my notes but i cannot find anything

Answer 7

you are wanting to know if \(A^3+I\) is invertible given that A is invertible...correct.

Answer 8

yes

Answer 9

then the matrix I gave above shows that \(A^3+I\) might not be invertible. (the zero matrix is not invertible)

Answer 10

oh ok. i worked it how using ur example n it doesn't have an inverse. correct me is im wrong....for A^3 +2A=I using the same matrix u provided it would also show t be false because the end matrix on the left and right would be different right?

Answer 11

A^3 +2A=I is false

Answer 12

yes...using the same A as above

Answer 13

thanks i really appreciate it your help i understand.