Question 1:
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
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:
Solution:
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
(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
m=1(2)(3)−(−5)(1)=111
k = 5
(b)