The image shows a cryptography-based problem where letters are replaced by numbers, and a system of matrices is used to decode a word that symbolizes basic elements like "air, water, earth, fire" (representing fundamental elements of the universe). The task asks to compute and find the word corresponding to these elements.

Matrix Equation Given:

$\begin{pmatrix} a & b \\ c & d \end{pmatrix} = (A + B)^T - C^T$ Where:

• $A = \begin{pmatrix} -3 & 4 \\ 3 & 0 \end{pmatrix}$
• $B = \begin{pmatrix} 2 & -1 \\ 4 & 8 \end{pmatrix}$
• $C = \begin{pmatrix} 2 & -1 \\ 10 & 3 \end{pmatrix}$

We need to:

1. Find $A + B$
2. Transpose $(A + B)$
3. Transpose $C$
4. Solve for $abcd$.

Step-by-Step Solution:

1. Sum of $A + B$: $A + B = \begin{pmatrix} -3 & 4 \\ 3 & 0 \end{pmatrix} + \begin{pmatrix} 2 & -1 \\ 4 & 8 \end{pmatrix} = \begin{pmatrix} -1 & 3 \\ 7 & 8 \end{pmatrix}$

2. Transpose of $(A + B)$: $(A + B)^T = \begin{pmatrix} -1 & 7 \\ 3 & 8 \end{pmatrix}$

3. Transpose of $C$: $C^T = \begin{pmatrix} 2 & 10 \\ -1 & 3 \end{pmatrix}$

4. Subtract $C^T$ from $(A + B)^T$: $(A + B)^T - C^T = \begin{pmatrix} -1 & 7 \\ 3 & 8 \end{pmatrix} - \begin{pmatrix} 2 & 10 \\ -1 & 3 \end{pmatrix} = \begin{pmatrix} -3 & -3 \\ 4 & 5 \end{pmatrix}$

So, we have: $\begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} -3 & -3 \\ 4 & 5 \end{pmatrix}$

Now, we map these values to letters using the cryptography table provided.

• $a = -3$ → F
• $b = -3$ → F
• $c = 4$ → M
• $d = 5$ → N

The letters are FFMN. The word could refer to something related to "air, water, earth, fire," but based on the cryptographic scheme, the exact word is FFMN.

Do you want more details or have any questions?

Related questions:

1. What other cryptographic methods could be used to decode words?
2. How can matrix transposition affect calculations in cryptography?
3. What role do matrices play in real-world cryptographic systems?
4. Could different mathematical operations replace the matrix operations here?
5. How does this cipher technique compare to classical substitution ciphers?

Tip:

When working with matrix equations in cryptography, keep track of transpositions and sign changes, as they significantly alter the final message.