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4.10 SPM Practice (Long Questions)

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Question 1:
It is given that matrix A = ( 3 − 1 5 − 2 )
(a) Find the inverse matrix of A.
(b) Write the following simultaneous linear equations as matrix equation:
3u – v = 9
5u – 2v = 13
Hence, using matrix method, calculate the value of u and v.

Solution:
(a)
A − 1 = 1 3 ( − 2 ) − ( 5 ) ( − 1 ) ( − 2 1 − 5 3 ) = − 1 ( − 2 1 − 5 3 ) = ( 2 − 1 5 − 3 )

(b)
( 3 − 1 5 − 2 ) ( u v ) = ( 9 13 ) ( u v ) = − 1 ( − 2 1 − 5 3 ) ( 9 13 ) ( u v ) = − 1 ( ( − 2 ) ( 9 ) + ( 1 ) ( 13 ) ( − 5 ) ( 9 ) + ( 3 ) ( 13 ) ) ( u v ) = − 1 ( − 5 − 6 ) ( u v ) = ( 5 6 ) ∴ u = 5 , v = 6


Question 2:
It is given that matrix A = ( 2 − 5 1 3 ) and matrix B = m ( 3 k − 1 2 ) such that AB = ( 1 0 0 1 )
(a) Find the value of m and of k.
(b) Write the following simultaneous linear equations as matrix equation:
2u – 5v = –15
u + 3v = –2
Hence, using matrix method, calculate the value of u and v.

Solution:
(a) Since AB = ( 1 0 0 1 ) , B is the inverse of A.

m = 1 ( 2 ) ( 3 ) − ( − 5 ) ( 1 ) = 1 11
k = 5

(b)
( 2 − 5 1 3 ) ( u v ) = ( − 15 − 2 ) ( u v ) = 1 11 ( 3 5 − 1 2 ) ( − 15 − 2 ) ( u v ) = 1 11 ( ( 3 ) ( − 15 ) + ( 5 ) ( − 2 ) ( − 1 ) ( − 15 ) + ( 2 ) ( − 2 ) ) ( u v ) = 1 11 ( − 55 11 ) ( u v ) = ( − 5 1 ) ∴ u = − 5 , v = 1