Find, in terms of m,det(A-mI) . This will be a quadratic expression in m.?

Question

The matrix A is given by 17 3
-90 -16

(i) What is this quadratic expression?
(ii) Find the larger solution of the equation det(A-mI) =0
(iii) Find the smaller solution of the equation det(A-mI) =0
(iv) Substitute the larger value of (found in part (ii)) into the matrix
(A-mI) .

Answer 1

(i) What is this quadratic expression?
(17-m)(-16-m) + 270 = m^2 - m - 272 + 270 = m^2 -m -2
(ii) Find the larger solution of the equation det(A-mI) =0
m^2 -m -2 =0
(m-2)(m+1) =0
so the larger solution is m=2.
(iii) Find the smaller solution of the equation det(A-mI) =0
in this case it is m=-1
(iv) Substitute the larger value of (found in part (ii)) into the matrix
(A-mI) .
15 ... 3
-90 -18 .

Answer 2

So you have this matrix A:

[17, 3
-90, -16]

And you're subtracting mI from it

m*
[1, 0
0, 1]

=

[m, 0
0, m]


A - mI =
[17-m, 3
-90, -16-m]

The determinate of that is:

(17-m)*(-16-m) - (3*-90)

That should get you started.