Theory of Intelligence
11TH•Mathematics•Matrices And Determinants•Question 11

Solution

Decode the message $\begin{bmatrix}11&25&22\\20&10&14\\43&41&41\end{bmatrix}$ that was encoded using matrix $A=\begin{bmatrix}1&1&-1\\1&0&1\\2&1&1\end{bmatrix}$, where the numbers 1 to 26 are corresponding to the letters A to Z, and 27 is representing space or "-".

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Solution

Given coded matrix:

C=[112522201014434141]C=\begin{bmatrix}11&25&22\\20&10&14\\43&41&41\end{bmatrix}

and encoding matrix:

A=[11−1101211]A=\begin{bmatrix}1&1&-1\\1&0&1\\2&1&1\end{bmatrix}

Decoding uses:

P=A−1CP=A^{-1}C

Compute A−1A^{-1}:

A−1=[12−1−1−32−1−11]A^{-1}=\begin{bmatrix}1&2&-1\\-1&-3&2\\-1&-1&1\end{bmatrix}

Now:

P=A−1C=[8491527181265]P=A^{-1}C= \begin{bmatrix}8&4&9\\15&27&18\\12&6&5\end{bmatrix}

Interpret numbers (1→A,…,26→Z1\to A,\dots,26\to Z, 27→27\to space):

  • Column 1: (8,15,12)→(H,O,L)(8,15,12)\to (H,O,L)
  • Column 2: (4,27,6)→(D,space,F)(4,27,6)\to (D,\text{space},F)
  • Column 3: (9,18,5)→(I,R,E)(9,18,5)\to (I,R,E)

So the decoded message is:

HOLD FIRE\boxed{\text{HOLD FIRE}}

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