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x  2y  5

3x  2y z  10

2x  4y z  13

 5 8  1 2
5. Given P    and Q    .
 3 1   1 5 
 1 3

(a) Determine matrix S if S  (P Q )R and R    .
 2 1 

(b) Hence, find the inverse of matrix S.

 2 2  2

6. If A    1  1 1 and XA = 4I where X is a 3 x 3 matrix and I is an identity matrix.

  8 0 4  

(a) Find the interest of matrix A using the adjoint method.

(b) Hence, find matrix X.

6   4
2
7. (a) The matrix A    . If A – pA – qI = 0 where p and q are real
 1 0 
numbers, I is the 2 x 2 identity matrix and 0 is the null matrix 2 x 2, find p and q.

1 1    3  x 2 


 

(b) Given that the matrix equation AX = B is 2 1  4 y   3 .



 

1 1 1      z  1  



(i) Find the determinant of matrix A.
 5 p 3 

(ii) Given the cofactor matrix of A    4 2  2 , find p and q.


 q  2 1 


(iii) Determine the adjoint matrix of A and hence find the inverse of A.