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4.10 SPM Practice (Long Questions)

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Question 3:
It is given that Q  ( 3 2 6 5 ) = ( 1 0 0 1 ) , 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 = ( 3 2 6 5 ) − 1 Q = 1 3 ( 5 ) − 2 ( 6 ) ( 5 − 2 − 6 3 ) Q = 1 3 ( 5 − 2 − 6 3 ) Q = ( 5 3 − 2 3 − 2 1 )

(b)
( 3 2 6 5 ) ( u v ) = ( 5 2 ) ( u v ) = 1 3 ( 5 − 2 − 6 3 ) ( 5 2 ) ( u v ) = 1 3 ( ( 5 ) ( 5 ) + ( − 2 ) ( 2 ) ( − 6 ) ( 5 ) + ( 3 ) ( 2 ) ) ( u v ) = 1 3 ( 21 − 24 ) ( u v ) = ( 7 − 8 ) ∴ u = 7 , v = − 8


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