Given `[["4 " " 2" ],[" -1 "" 1 " ]]` M = 6I , where M is a matrix and I is unit matrix of order 2×2.
(i) State the order of matrix M.
(ii) Find the matrix M.
Solution
(i) `[["4 " " 2" ],[" -1 "" 1 " ]] M = 6I , 6 [[ "1 " " 0" ],[" 0 " " 1 " ]]`
`[["4 " " 2" ],[" -1 "" 1 " ]] M = [[ "6 " " 0" ],[" 0 " " 6 " ]]`
∴ M has the order of 2 × 2
(ii) Let us assume the matrix M = `[["a " " b" ],[" c " " d "]]`
`[["4 " " 2" ],[" -1 "" 1 " ]] [["a " " b" ],[" c " " d "]] = [[ "6 " " 0" ],[" 0 " " 6 " ]]`
⇒ `[["4a + 2c " " 4b + 2d" ],["-a + c " " -b + d" ]] = [[ "6 " " 0" ],[" 0 " " 6 " ]]`
⇒ -a + c = 0 ⇒ a = c
4b + 2d = 0
⇒ 4b = -2d
d = - 2b
Now , 4a + 2c = 6 ⇒ 4a + 2a = 6 ⇒ 6a= 6 ⇒ a = 1 - C
also , - b + d = 6
- b + (-2b) = 6
- 3b = 6
b = -2
∴ a = 1 , b = -2 , c = 1 , d = 4
