00:01
Hi, in this question we are given with the matrix a that is equals to 32 minus 70 minus 140 5 minus 13 minus 20 5 minus 10 minus 23 and we need to find the invertible matrix p and the diagonal matrix d.
00:29
So we will start with finding the characteristic polynomial that is determinant for a minus lambda i that would be equals to determinant for 32 minus lambda minus 70 minus 140 5 minus 13 minus lambda minus 1 .40.
00:52
5 minus 13 minus lambda minus 20 5 minus 10 minus 23 minus lambda minus lambda.
01:00
Solving this further we get the result to be equals to minus lambda cube minus four lambda square plus three lambda plus 18 and solving this further we can write this as minus solving this further we can write this to be equals to minus lambda minus two times lambda plus three whole square from here we get the values for lambda that is equals to 2 minus 3 and minus 3.
01:42
First find the eigenvector associated to the eigenvalue lambda as equals to.
01:51
So we get a minus 2i to be equals to 30 minus 70 minus 140, 5 minus 13 minus 13 minus 13 minus 20 minus 20, 5 minus 13 minus 20, 5 minus 10 minus 23.
02:18
Solving this further and writing the reduced row echlin form of this matrix, reduced row echlin form will be 1 .0 minus 7.
02:37
0 .1 minus 1 .000.
02:42
To find the null space, we will solve the matrix.
02:49
As this reduced row ethylene form of matrix, 1 -0 -7 -0 -1 -1 -1 -0 -0 -0 -0 -t, x -1 -t, x -3 to be equals to 0 -0 -0...