Matrices Long Questions (Question 1 - 4)


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:
3uv = 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


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