1 Find the eigenvalues and a basis of the eigenspace for eac

1. Find the eigenvalues and a basis of the eigenspace for each eigenvalue of A .

2. Find the eigenvalues and a basis of the eigenspace for each eigenvalue of B .

3. Find a matrix P so that A = P^-1BP or show why it does not exist

Finding eigen values for matrix A

we substitute A into the equation det(A-kI)=0 and solve for k

|A-kI| = 0 eq. 1

where I is identity matrix

On solving equation eq.1

(-3-k)*((-k)*(1-k)- (-2)*(-1)) - (-5)*((-2)*(1-k) - (-2)*(-1)) + 5*((-2)*(-2) - (-2)*(-k)) = 0

solve for k and find 3 values of above equation k1,k2&k3 are eigen values for matrix A

Finding eigen values for matrix B

we substitute B into the equation det(B-mI)=0 and solve for m

|B-mI| = 0 eq. 2

where I is identity matrix

solve for m and find 3 values of above equation m1,m2&m3 are eigen values for matrix B

For finding basis of each matrix simplify the matrix in unit 3*3 matrix