4.10 Matrices, SPM Paper 2 (Long Questions)


Question 5:
(a) Given  1 14 ( 2 s 4 t )( t 1 4 2 )=( 1 0 0 1 ), find the value of s and of t.
(b) Write the following simultaneous linear equations as matrix form:
3x – 2y = 5
9x + y = 1
Hence, using matrix method, calculate the value of x and y.

Solution:
(a) 1 14 ( 2 s 4 t )( t 1 4 2 )=( 1 0 0 1 ) 1 14 ( 2t+4s 2+2s 4t+4t 4+2t )=( 1 0 0 1 ) 2+2s 14 =0   2s=2      s=1 4+2t 14 =1 4+2t=14 2t=10 t=5

(b) ( 3 2 9 1 )( x y )=( 5 1 )   ( x y )= 1 21 ( 1 2 9 3 )( 5 1 )   ( x y )= 1 21 ( ( 1 )( 5 )+( 2 )( 1 ) ( 9 )( 5 )+( 3 )( 1 ) )   ( x y )= 1 21 ( 7 42 )   ( x y )=( 1 3 2 ) x= 1 3 ,  y=2


Question 6:
It is given that matrix P=( 6 3 5 2 ) and matrix Q= 1 m ( 2 3 5 n )  such that PQ=( 1 0 0 1 ).
(a) Find the value of m and of n.
(b) Write the following simultaneous linear equations as matrix form:
6x – 3y = –24
–5x + 2y = 18
Hence, using matrix method, calculate the value of x and y.

Solution:
(a) m=6( 2 )( 3 )( 5 )   =1215 m=3 n=6

(b) ( 6 3 5 2 )( x y )=( 24 18 )   ( x y )= 1 1215 ( 2 3 5 6 )( 24 18 )   ( x y )= 1 3 ( ( 2 )( 24 )+( 3 )( 18 ) ( 5 )( 24 )+( 6 )( 18 ) )   ( x y )= 1 3 ( 6 12 )   ( x y )=( 2 4 ) x=2,  y=4


Question 7:
(a) Find the inverse matrix of ( 3 2 5 4 ).
(b) Ethan and Rahman went to the supermarket to buy cucumbers and carrots. Ethan bought 3 cucumbers and 2 carrots for RM9. Rahman bought 5 cucumbers and 4 carrots for RM16.
By using matrix method, find the price, in RM, of a cucumber and the price of a carrot. 
Solution:
(a) Inverse matrix of ( 3 2 5 4 ) = 1 1210 ( 4 2 5 3 ) = 1 2 ( 4 2 5 3 ) =( 2 1 5 2 3 2 )

(b) 3x+2y=9……………..( 1 ) 5x+4y=16……………( 2 ) ( 3 2 5 4 )( x y )=( 9 16 )               ( x y )=( 2 1 5 2 3 2 )( 9 16 )               ( x y )=( ( 2 )( 9 )+( 1 )( 16 ) ( 5 2 )( 9 )+( 3 2 )( 16 ) )               ( x y )=( 1816 45 2 +24 )               ( x y )=( 2 3 2 ) x=2,  y= 3 2 Price of a cucumber=RM2    Price of a carrot=RM1.50

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