Question 8.4.7: Decoding a Word Decode -29, 43, -64, 92 from Example 6.

Step 1 Find the inverse of the coding matrix. The coding matrix in Example 6 was \left[\begin{array}{rr}-2 & -3 \\ 3 & 4\end{array}\right]. We use the formula for the multiplicative inverse of a 2 × 2 matrix to find the multiplicative inverse of this matrix. It is \left[\begin{array}{rr}4 & 3 \\ -3 & -2\end{array}\right].

Step 2 Multiply the multiplicative inverse of the coding matrix and the coded matrix.

Step 3 Express the numbers, by columns, from the matrix in step 2 as letters. The numbers are 13, 1, 20, and 8. Using letters, the decoded message is MATH.