4.10 SPM Practice (Long Questions)


Question 3:
It is given that Q  (3265)=(1001) , where Q is a 2 x 2 matrix.
(a) Find Q.
(b) Write the following simultaneous linear equations as matrix equation:
3u + 2v = 5
6u + 5v = 2
Hence, using matrix method, calculate the value of u and v.

Solution
:

(a) 
Q=(3265)1Q=13(5)2(6)(5263)Q=13(5263)Q=(532321)

(b)

(3265)(uv)=(52)(uv)=13(5263)(52)(uv)=13((5)(5)+(2)(2)(6)(5)+(3)(2))(uv)=13(2124)(uv)=(78)


Question 4:
It is given that Q ( 3 2 5 4 ) = ( 1 0 0 1 )  , where Q is a 2 × 2 matrix.
(a) Find the matrix Q.
(b) Write the following simultaneous linear equations as matrix equation:
3x – 2y = 7
5x – 4y = 9
Hence, using matrix method, calculate the value of x and y.

Solution:

(a)
Q = 1 3 ( 4 ) ( 5 ) ( 2 ) ( 4 2 5 3 ) = 1 2 ( 4 2 5 3 ) = ( 2 1 5 2 3 2 )

(b)

( 3 2 5 4 ) ( x y ) = ( 7 9 ) ( x y ) = 1 2 ( 4 2 5 3 ) ( 7 9 ) ( x y ) = 1 2 ( ( 4 ) ( 7 ) + ( 2 ) ( 9 ) ( 5 ) ( 7 ) + ( 3 ) ( 9 ) ) ( x y ) = 1 2 ( 10 8 ) ( x y ) = ( 5 4 ) x = 5 , y = 4