Question

Show that the matrix [ 4 6 6 1 −1 3 2 −5 −2 ] is not diagonalizable

Answer 1

$$\begin{pmatrix} 4& 6 & 6 \\ 1 & -1 & 3 \\ 2 & -5 & -2 \\ \end{pmatrix}$$

Start from forming a new matrix by subtracting $\lambda$  from the diagonal entries of the given matrix:

$$\begin{pmatrix} 4-\lambda & 6 & 6 \\ 1 & -1-\lambda & 3 \\ 2 & -5 & -2-\lambda \\ \end{pmatrix}$$

$$\begin{vmatrix} 4-\lambda & 6 & 6 \\ 1 & -1-\lambda & 3 \\ 2 & -5 & -2-\lambda \\ \end{vmatrix}$$

$$=(4-\lambda)\begin{vmatrix} -1-\lambda & 3 \\ -5 & -2-\lambda \end{vmatrix}-6\begin{vmatrix} 1 & 3 \\ 2 & -2-\lambda \end{vmatrix}$$

$$+6\begin{vmatrix} 1 & -1-\lambda \\ 2 & -5 \end{vmatrix}$$

$$=(4-\lambda)(1+\lambda)(2+\lambda()+15(4-\lambda)$$

$$+6(2+\lambda)+36-30+12(1+\lambda)$$

$$=8+12\lambda+4\lambda^2-2\lambda-3\lambda^2-\lambda^3+60-15\lambda$$

$$+12+6\lambda+6+12+12\lambda$$

$$=-\lambda^3+\lambda^2+13\lambda+98$$

Solve the equation 

$$-\lambda^3+\lambda^2+13\lambda+98=0$$

Only approximate roots can be found.

The roots are 

$\lambda_1\approx5.9513233156912$

$\lambda_2\approx−2.4756616578456+3.21527996418223i$

$\lambda_3\approx−2.4756616578456-3.21527996418223i$

Since there are complex roots, the matrix is not diagonalizable.