Can you please help me solve this linear algebra problem?

5. Triangularisation with an orthogonal matrix Example 7.9 in the Study Guide (pages 2123 of Topic 7) shows the triangularisation procedure for a matrix. Consider the following matrix A, which also has eigenvalues 1, 1 and 5. 3 2 2 A: 1 2 1 1 1 2 i. Construct a matrix S such that it is an orthogonal matrix with the rst column corresponding 1 with the eigenvector 0 . Calculate 81118. 1 1 bT 0 A1 1 and 5 by determining and then simplifying its characteristic equation. ii. Show that for the resulting matrix, S'lAS = [ 1 , the 2 X 2 matrix A1 has eigenvalues iii. Find the eigenvector from A1 corresponding with A = 1 and then construct an orthogonal 2 X 2 matrix Q where the rst column is based on your eigenvector. Hence construct the matrix 100 R=U 0Q iv. Calculate P = SR. Show that P is an orthogonal matrix and verify that P'lAP is upper triangular. 5 marks