Question

Give an example, with explanation, of a matrix with integer

entries, whose characteristic polynomial is x3-x2+x-1.

Answer 1

To find a matrix with a given characteristic polynomial, we start by considering the coefficients of the polynomial. The companion matrix for this polynomial is [0 0 1; 1 0 -1; 0 1 -(-1)].

Step-by-step explanation:

Step-by-step explanation:

To find a matrix with a given characteristic polynomial, we start by considering the coefficients of the polynomial.

The characteristic polynomial x^3-x^2+x-1 has coefficients: a3 = 1, a2 = -1, a1 = 1, a0 = -1.

We can construct a matrix by considering the companion matrix, which is a matrix with the characteristic polynomial as its characteristic equation. The companion matrix for this polynomial is:

[0 0 -a0;
1 0 -a1;
0 1 -a2]

Substituting in the coefficients, we get the following matrix:

[0 0 1;
1 0 -1;
0 1 -(-1)]

Answer:

The matrix with integer entries whose characteristic polynomial is x^3-x^2+x-1 is:

[0 0 1;
1 0 -1;
0 1 1]