Question

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.

Answer 1

(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