What is the inverse of the 3x3 matrix mod 26

[DODGEVIPER13]

Homework Statement

What is the inverse of the 3x3 matrix mod 26?
K = $$\begin{pmatrix} 17 & 17 & 5\\ 21 & 18 & 21\\ 2 & 2 & 19 \end{pmatrix}$$

Homework Equations

The Attempt at a Solution

So I found all the cofactors and then took the transpose of the matrix. I then divided new matrix, by the determinate -939. After which I would multiply this by 17 because 23^-1 mod 26 = 17 to get the inverse. I found 17 by using the euclidean algorithm. This was all UPLOADED. However I am confused because even if I do this I do not get the answer in the book. They get:

$$\begin{pmatrix} 4 & 9 & 15\\ 15 & 17 & 6\\ 24 & 0 & 17 \end{pmatrix}$$

I have so far without multiplying it by 17:

$$\begin{pmatrix} 300/-939 & -313/-939 & 267/-939\\ -357/-939 & 313/-939 & -252/-939\\ 6/-939 & 0 & -51/-939 \end{pmatrix}$$

I realize that even if I go ahead I will not reach what the book has, what have I done wrong? All of my work has been UPLOADED.

 

[.Scott]

I'll do the center column - because it's easy:
$$\begin{pmatrix} k_{11} & k_{12} & k_{13}\\ k_{21} & k_{22} & k_{23}\\ k_{31} & k_{32} & k_{33} \end{pmatrix} \begin{pmatrix} 17 & 17 & 5\\ 21 & 18 & 21\\ 2 & 2 & 19 \end{pmatrix} = \begin{pmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{pmatrix} \ \\ \ \\ \ \\ A)\ 17K_{11} + 21K_{12} + 2K_{13} = 1 \\ B)\ 17K_{11} + 18K_{12} + 2K_{13} = 0 \\ C)\ 5K_{11} + 21K_{12} + 19K_{13} = 0 \\ D)\ 17K_{21} + 21K_{22} + 2K_{23} = 0 \\ E)\ 17K_{21} + 18K_{22} + 2K_{23} = 1 \\ F)\ 5K_{21} + 21K_{22} + 19K_{23} = 0 \\ G)\ 17K_{31} + 21K_{32} + 2K_{33} = 0 \\ H)\ 17K_{31} + 18K_{32} + 2K_{33} = 0 \\ I)\ 5K_{31} + 21K_{32} + 19K_{33} = 1 \\ \ \\ \ \\ Modulo 26\ table\ for\ Y=3X: 0, 3, 6, 9, 12, 15, 18, 21, 24, 1, 4, 7, 10, 13, 16, 19, 22, 25, ...\\ A-B)\ \ \ 3K_{12} = 1 \\ A-B/3)\ K_{12} = 9 \\ D-E)\ \ \ 3K_{22} = 25 \\ D-E/3)\ K_{22} = 17 \\ G-H)\ \ \ 3K_{32} = 0 \\ G-H/3)\ 3K_{32} = 0 \\ \ \\ \ \\ \begin{pmatrix} k_{11} & 9 & k_{13}\\ k_{21} & 17 & k_{23}\\ k_{31} & 0 & k_{33} \end{pmatrix} \ \\$$
That book answer is looking good to me.

 

[.Scott]

The Attempt at a Solution

So I found all the cofactors and then took the transpose of the matrix. I then divided new matrix, by the determinate -939. After which I would multiply this by 17 because 23^-1 mod 26 = 17 to get the inverse. I found 17 by using the euclidean algorithm. This was all UPLOADED. However I am confused because even if I do this I do not get the answer in the book. They get:

$$\begin{pmatrix} 4 & 9 & 15\\ 15 & 17 & 6\\ 24 & 0 & 17 \end{pmatrix}$$

I have so far without multiplying it by 17:

$$\begin{pmatrix} 300/-939 & -313/-939 & 267/-939\\ -357/-939 & 313/-939 & -252/-939\\ 6/-939 & 0 & -51/-939 \end{pmatrix}$$

I realize that even if I go ahead I will not reach what the book has, what have I done wrong? All of my work has been UPLOADED.

Mod 26 (-939) is 23.
Mod 26 (1/23) is 17.
So 17 is the right number to use.

$$\begin{pmatrix} 17*300 & 17*-313 & 17*267\\ 17*-357 & 17*313 & 17*-252\\ 17*6 & 0 & 17*-51 \end{pmatrix} = \begin{pmatrix} 17*14 & 17*25 & 17*7\\ 17*7 & 17*1 & 17*8\\ 17*6 & 0 & 17*1 \end{pmatrix} = \begin{pmatrix} 4 & 9 & 15\\ 15 & 17 & 6\\ 24 & 0 & 17 \end{pmatrix}$$
So you already had the solution.
If you're using a recent Windows operating system, you have a calculator with the Mod function.

 

[DODGEVIPER13]

Ok cool so I did get the answer.

 

[rezeew]

Your approach to finding the inverse of the matrix is correct. However, there may be a mistake in your calculation of the determinant. The determinant of the given matrix is -939 mod 26 = 17, not -939. This could be the reason why your final answer does not match the one in the book.

Also, be careful with the use of negative numbers in modular arithmetic. Instead of -313, it should be 16 (since 313 mod 26 = 16). Similarly, instead of -357, it should be 20, and instead of -252, it should be 22. After correcting these errors, you should get the same answer as the one in the book.

Another approach to finding the inverse of a matrix mod 26 is to use the extended Euclidean algorithm. This algorithm will give you the inverse directly without having to calculate the cofactors and transpose the matrix. You can try using this method to check your answer as well.