Matrix diagonalizability

Question

Consider the 4x4 matrix

A = [1 2 4 0

0 -1 1 7

0 0 0 6

0 0 0 7]

Is this matrix diagonalizable? Why or why not.

Answer 1

1 2 4 0

0 -1 1 7

0 0 0 6

0 0 0 7

-row3 + row4

1 2 4 0

0 -1 1 7

0 0 0 6

0 0 0 1

- 6 * row4 + row3

1 2 4 0

0 -1 1 7

0 0 0 0

0 0 0 1

swaps row 3 and row 4

1 2 4 0

0 -1 1 7

0 0 0 1

0 0 0 0

-7*row3 + row2

1 2 4 0

0 -1 1 0

0 0 0 1

0 0 0 0

2 * row2 + row1

-4 * row 2 + row 1

1 0 0 0

0 -1 1 0

0 0 0 1

0 0 0 0

-row2

1 0 0 0

0 1 -1 0

0 0 0 1

0 0 0 0

see the problem?