# Thread: show for a if a square matrix A...it will hold for A^T

1. ## show for a if a square matrix A...it will hold for A^T

Show that if for a square matrix $A$ satisfies $A^3+ 4A^2 - 2A + 7I = 0$ then so does $A^T$

I suspect that this is not always true so I need to do an example not the general case ( I partially suspect this cause the next question asks me if this is always true or sometimes false so I need to find a counter-example as well)

not really sure where to start this problem.

2. ## Re: show for a if a square matrix A...it will hold for A^T

Originally Posted by Jonroberts74
Show that if for a square matrix $A$ satisfies $A^3+ 4A^2 - 2A + 7I = 0$ then so does $A^T$

I suspect that this is not always true so I need to do an example not the general case ( I partially suspect this cause the next question asks me if this is always true or sometimes false so I need to find a counter-example as well)

not really sure where to start this problem.
$\left(A^T\right)^2=\left(A^2\right)^T$

$\left(A^T\right)^3=\left(A^3\right)^T$

$(A^3+4A^2-2A+7I)^T=0^T=0$

$(A^3)^T+(4A^2)^T-(2A)^T+(7I)^T=0$

$(A^T)^3+4(A^T)^2-2(A^T)+7I=0$

so $A^T$ satisfies the equation as well.

3. ## Re: show for a if a square matrix A...it will hold for A^T

ah okay, in my notes I have $(A^{-1})^T = (A^T)^{-1}$ but I didn't know it worked for powers, not just inverses. So then it is always true that if a matrix A satisfies that equation then $A^T$ will as well?

Does A have to be symmetric for this to work?

4. ## Re: show for a if a square matrix A...it will hold for A^T

Originally Posted by Jonroberts74
ah okay, in my notes I have $(A^{-1})^T = (A^T)^{-1}$ but I didn't know it worked for powers, not just inverses. So then it is always true that if a matrix A satisfies that equation then $A^T$ will as well?

Does A have to be symmetric for this to work?
You know that $\displaystyle \left(AB\right)^T = B^TA^T$. Just set $\displaystyle A = B$.

5. ## Re: show for a if a square matrix A...it will hold for A^T

Originally Posted by Jonroberts74
ah okay, in my notes I have $(A^{-1})^T = (A^T)^{-1}$ but I didn't know it worked for powers, not just inverses. So then it is always true that if a matrix A satisfies that equation then $A^T$ will as well?

Does A have to be symmetric for this to work?
no

$(A\cdot A \cdot A \dots)^T=A^T\cdot A^T \cdot A^T \dots$

so

$(A^n)^T=(A^T)^n$