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2.10.1 Matrices, SPM Paper (Long Questions)


Question 1:
It is given that matrix A = (3−15−2)
(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=13(−2)−(5)(−1)(−21−53)=−1(−21−53)=(2−15−3)

(b)
(3−15−2)(uv)=(913)(uv)=−1(−21−53)(913)(uv)=−1((−2)(9)+(1)(13)(−5)(9)+(3)(13))(uv)=−1(−5−6)(uv)=(56)∴u=5,v=6


Question 2:
It is given that matrix A = (2−513) and matrix B = m(3k−12) such that AB = (1001)
(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 = (1001) , B is the inverse of A.

m=1(2)(3)−(−5)(1)=111
k = 5

(b)
(2−513)(uv)=(−15−2)(uv)=111(35−12)(−15−2)(uv)=111((3)(−15)+(5)(−2)(−1)(−15)+(2)(−2))(uv)=111(−5511)(uv)=(−51)∴u=−5,v=1



Question 3:
It is given that Q  (3265)=(1001) , 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=(3265)−1Q=13(5)−2(6)(5−2−63)Q=13(5−2−63)Q=(53−23−21)

(b)
(3265)(uv)=(52)(uv)=13(5−2−63)(52)(uv)=13((5)(5)+(−2)(2)(−6)(5)+(3)(2))(uv)=13(21−24)(uv)=(7−8)∴u=7,v=−8


Question 4:
It is given that Q(3−25−4)=(1001)  , 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=13(−4)−(5)(−2)(−42−53)=−12(−42−53)=(2−152−32)

(b)
(3−25−4)(xy)=(79)(xy)=−12(−42−53)(79)(xy)=−12((−4)(7)+(2)(9)(−5)(7)+(3)(9))(xy)=−12(−10−8)(xy)=(54)∴x=5,y=4

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