2.10.3 Matrices, SPM Paper 2 (Long Questions)


Question 8:
The inverse matrix of   ( 4 −1 2 5 ) is t( 5 1 −2 n ).
(a) Find the value of n and of t.
(b) Write the following simultaneous linear equations as matrix equation:
4x – y = 7
2x + 5y = –2
Hence, using matrix method, calculate the value of x and of y.


Solution:
(a) t( 5 1 −2 n )= ( 4 −1 2 5 ) −1 = 1 ( 4 )( 5 )−( −1 )( 2 ) ( 5 1 −2 4 ) = 1 22 ( 5 1 −2 4 ) ∴t= 1 22 , n=4

(b) ( 4 −1 2 5 )( x y )=( 7 −2 )   ( x y )= 1 22 ( 5 1 −2 4 )( 7 −2 )   ( x y )= 1 22 ( ( 5 )( 7 )+1( −2 ) ( −2 )( 7 )+( 4 )( −2 ) )   ( x y )= 1 22 ( 35−2 −14−8 )   ( x y )= 1 22 (  33 −22 )   ( x y )=(   3 2 −1 ) ∴x= 3 2 ,  y=−1


Question 9:
 (a) Given  1 s ( −4 2 −5 3 )( t −2 5 −4 )=( 1 0 0 1 ), find the value of s and of t.

(b) Using matrices, calculate the value of x and of y that satisfy the following matrix equation:
( −4 2 −5 3 )( x y )=( 1 2 )


Solution:
(a) 1 s ( t −2 5 −4 )= ( −4 2 −5 3 ) −1 = 1 ( −4 )( 3 )−( 2 )( −5 ) ( 3 −2 5 −4 ) = 1 −2 ( 3 −2 5 −4 ) ∴s=−2, t=3

(b) ( −4 2 −5 3 )( x y )=( 1 2 )   ( x y )= 1 −2 ( 3 −2 5 −4 )( 1 2 )   ( x y )= 1 −2 ( ( 3 )( 1 )+( −2 )( 2 ) ( 5 )( 1 )+( −4 )( 2 ) )   ( x y )= 1 −2 ( −1 −3 )   ( x y )=( 1 2 3 2 ) ∴x= 1 2 ,  y= 3 2

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