Question

Using the matrix A = `[[-1,2],[-1,1]]`decode the message of matrix C= `[[4,11,12,-2],[-4,4,9,-2]]`

Answer 1

Encoding Matrix A = ` [[-1,2],[-1,1]]` 

Given, C = `[[4,11,12,2],[-4,4,9,-2]]`

Step 1:
Writing the numbers in C matrix column wise gives the encoded message. 

∴ Encoded Message = 4 -4 11 4 12 9 -2 -2
This Encoded message is transmitted.
Assume there is no corruption of data, the message at the receiving end is 4 -4 11 4 12 9 -2 -2
This message is decoded
Step 2;
We know, if P = `[[a,b],[c,d]]` then `p_1= 1/|p| [[d,-b],[-c,a]]`

From (1), |A| = -1 + 2 = 1                     ................(2)

∴Decoding matrix `A_1=1/1 [[1,-2],[1,-1]]` (From 2)    ........(3)

From (2) & (3), `A^-1C=[[1,-2],[1,-1]]xx [[4,11,12,-2],[-4,4, 9,-2]]`

∴ `A^-1C=[[4+8,11-8,12-18,-2+4],[4+4,11-4,12-9,-2+2]]`

∴ `A^-1 C=[[12,3,6,2],[8,7,3,0]]`

Step 3:
Considering the numbers column-wise we get,
12 8 3 7 -6 3 2 0 

Dcoded Message = `12  8  3  7 -6  3  2  0 or [[12,3,6,2],[8,7,3,0]]`