Show that the two matrices `A, P^(-1)AP` have the same characteristic roots

If A and P are the square matrices of the same order and if P be invertible, show that the matrices A and P^(-1) have the same characteristic roots.

For any two matrices A and B , we have

P is a non-singular matrix and A, B are two matrices such that B=P^(-1) AP . The true statements among the following are

Show that If the product of two matrices is a zero matrix, it is not necessary that one of the matrices is a zero matrix.

If x^(2)+px+q=0andx^(2)+qx+p=0,(p!=q) have a common roots,show that their other 1+p+q=0 .Also,show that their other roots are the roots of the equation x^(2)+x+pq=0.

The sphere and cube have same surface. Show that the ratio of the volume of sphere to that of cube is root 6 : root pi

If A and B are any two square matrices of the same order than ,

Two A.P.'s have the same common difference. The first term of one A.P. is 2 and that of the other is 7. Show that the difference between their 10th terms is the same as the difference between their 21th terms, which is the same as the difference between any two corresponding terms.

Show that the characteristics roots of an idempotent matris are either 0 or 1