Diagonalize the following matrix 74 0 4 0 4 14 0 0 3 0 0 0 0

Diagonalize the following matrix. [7-4 0 4 0 4 1-4 0 0 3 0 0 0 0 3] Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. For P =, D = [3 0 0 0 0 3 0 0 0 0 4 0 0 0 0 7] B. The matrix cannot be diagonalized.

Solution

Let the given matrix be denoted by A. From the description of D, we observe that the eigenvalues of A are 7,4,3,3. Further, we know that the eigenvector(s) of A corresponding to its eigenvalue are solutions to the equation (A- I₄)X = 0. When =3, we will reduce A-3I₄ to its RREF as under:

Multiply the 1st row by ¼

Add 1 times the 2nd row to the 1st row

Then, the RREF of A-3I₄ is

1

0

1

-3

0

1

1

-4

0

0

0

0

0

0

0

0

Now, if X = (x,y,z,w)^T, then the above equation is equivalent to x+z-3w=0 and y+z-4w = 0. Then X =                 (-z+3w,-z+4w,z,w)^T = z(-1,-1,1,0)^T + w(3,4,0,1)^T. Hence, the eigenvectors of A corresponding to its eigenvalue 3 are (-1,-1,1,0)^T and(3,4,0,1)^T. Similarly, the eigenvectors of A corresponding to its eigenvalue 4 and 7 are (4,3,0,0)^T and (1,0,0,0)^T respectively. Apparently, A has 4 distinct linearly independent eigenvectors.Thus A is diagonalizable and P =

-1

3

4

1

-1

4

3

0

1

0

0

0

0

1

0

0

1	0	1	-3
0	1	1	-4
0	0	0	0
0	0	0	0