Maths Di

Stage II Maths Studies

Suppose all letters are given a number from 1 to 26. E.g. a=1, b=2, c=3 etc then “put out the cat” becomes 16 21 20 15 21 20 20 8 5 3 1 20. If we use the following randomly chosen matrix

A= [■(14&8&3@8&5&2@3&2&1)]

as a multiplier for each of these then

‘put’ = B = [16 21 20] becomes [452 273 110]

Verify this and ascertain the code for “out”, “the”, and “cat”. How can this message be coded by one matrix calculation?

PUT = [16 21 20] x [■(14&8&3@8&5&2@3&2&1)]

= [(16 x 14+21 x 8+20 x 3)(16 x 8+21 x 5+20 x 2)(16 x 3+21 x 2+20 x 1)]

= [ (224+168+60) (128+105+40) (48+42+20) ]

= [452 273 110]

OUT = [15 21 20] x [■(14&8&3@8&5&2@3&2&1)]

= [(15 x 14+21 x 8+20 x 3)(15 x 8+21 x 5+20 x 2)(15 x 3+21 x 2+20 x 1)]

= [ (210+168+60) (120+105+40) (45+42+20) ]

= [438 265 107]

THE = [20 8 5] x [■(14&8&3@8&5&2@3&2&1)]

= [(20 x 14+8 x 8+5 x 3)(20 x 8+8 x 5+5 x 2)(20x 3+8 x 2+5 x 1)]

= [ (280+64+15) (160+40+10) (60+16+5) ]

= [359 210 81]

CAT = [3 1 20] x [■(14&8&3@8&5&2@3&2&1)]

= [(3 x 14+1 x 8+20 x 3)(3 x 8+1 x 5+20 x 2)(3x 3+1 x 2+20 x 1)]

= [ (42+8+60) (24+5+40) (9+2+20) ]

= [110 69 31]

This message can be coded by one matrix calculation by putting it vertically in a 3 x 4 matrix and then multiplying it by the randomly chosen matrix like so:

[■(16&15&20 3@21&21&8 1@20&15&5 20)]

[■(14&8&3@8&5&2@3&2&1)] X [■(16&15&20 3@21&21&8 1@20&15&5 20)] = [■(452&438&359 110@273&265&210 69@110&107&81 31)]

Are there similar codes for the same letters?

U = 273, 265

T = 110, 107, 359, 31

No, there are no similar codes for the same letters as U equals to 273 and 265, while T equals to 110, 107, 359 and 31.

Compute the inverse of A and use it to decode your coded message.

According to the TI-84 Plus Texas Instruments Graphics Calculator:

A-1 = [■(1&-2&1@-2&5&-4@1&-4&6)]

[■(1&-2&1@-2&5&-4@1&-4&6)] X [■(452&438&359...