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Assignment 6   -   Solutions

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Use Cramer’s rule to solve the linear system
       
and also solve the system using the inverse A^-1 of the coefficient matrix A.

Use Cramer’s rule to solve the linear system
       
and verify your answer by substituting it into the left side of the linear system.

OR   use a cofactor expansion down column 1:

For   A₁   use a cofactor expansion along row 3:

For   A₂   use a cofactor expansion down column 1:

For   A₃   use a cofactor expansion along row 3:

Verification of this solution:
   
   
   

Find the adjugate [the transpose of the matrix of cofactors]
and hence the inverse   (A^-1)   of the matrix
       
and use this inverse matrix to verify your answer to question (2) above.

The sub-matrix   A_ij   is what remains of matrix   A   after the deletion of row i and column j.
The cofactors are  






From question 2,   det A = 88

Find the eigenvalues and corresponding set of basic eigenvectors for

Write down the matrix   P   that diagonalizes   A
and verify by matrix multiplication that   P ^-1AP = D.
Hence find   A ⁴³.

l = –1:

The (–1)-eigenvector is therefore any non-zero multiple of  

l = +1:

The (1)-eigenvector is therefore any non-zero multiple of  

Find the eigenvalues and corresponding set of basic eigenvectors for
       
Write down the matrix   P   that diagonalizes   A .

The matrix   A   is upper triangular  
All eigenvalues have multiplicity 1, which guarantees the existence of a matrix   P   that diagonalizes   A

l = –1:

The (–1)-eigenvector is therefore any non-zero multiple of  

l = +1:

The (1)-eigenvector is therefore any non-zero multiple of  

l = +2:

The (2)-eigenvector is therefore any non-zero multiple of  

Therefore the matrix   P   that diagonalizes   A   is

and the set of basic eigenvectors is just the three columns of   P .