Inverse of matrix mod $26$ wolframalpha wrong

Question

I want to find $A^{-1} \pmod{26}$ for

$A=\begin{bmatrix}10&3\\5&3\end{bmatrix}$

and I did the conventional $\frac{1}{ad-bc}\begin{bmatrix}d&-b\\-c&a\end{bmatrix}$ and found the inverse of the fraction mod $26$, cool, then reduced mod $26$ the matrix.

I obtained:

$$\begin{bmatrix}21&5\\17&18\end{bmatrix}$$

Cool - but the wolframalpha calculator obtained the transpose of above? What the?

Who is right?

Answer 1

Be careful that Wolfram addresses your $a,b,c,d$ to $1,2,3,4$, so you should have written your $a,b,c,d$ in the corresponding order to the entries $1,2,3,4$, which gives the matrix $$ \begin{bmatrix} a & b \\ c & d \\ \end{bmatrix} $$ It is not handy, but that is how this beta widget works.