# How to find the inverse of the matrix over $\mathbb Z_5$

How to find the inverse of the matrix over $\mathbb Z_5$

$$\left( \begin{matrix} 1 & 2& 0\\ 0 &2& 4 \\ 0& 0& 3\\ \end {matrix} \right)$$

• I know how to get inverse by adj / det operation.I am not familiar about solving this kind of questions.How should I approach by taking 9 variables matrix and solving the equations? I am confused. – jolkol piu Jan 23 '15 at 6:51
• In that case, compute the inverse by the adj/det operation. Where do you get stuck trying to do so? – anomaly Jan 23 '15 at 6:52
• When you compute the inverse normally, just figure out what each element is mod 5, yes? – Alfred Yerger Jan 23 '15 at 7:00
• Note that the determinant is $1 \cdot 2 \cdot 3 = 6 = 1\!\!\!\! \mod\!\! 5$ so the inverse in this case is the adjoint matrix, since $1/\text{det} = 1/1 = 1$ ( all $\!\!\!\!\mod 5$ ). – orangeskid Jan 23 '15 at 7:28
• You can find several similar questions on the site (for example, have a look at the list of related question on the right). Just randomly picking two similar questions: math.stackexchange.com/questions/280522/… and math.stackexchange.com/questions/85753/… – Martin Sleziak Jan 23 '15 at 8:11

Hint: One can proceed as usual by taking the given matrix $A$ and row-reducing the adjoined matrix $[\,A\,|\,\mathbb{I}\,]$. Note that the matrix is upper triangular, which makes available a very efficient method, because it implies the following about the inverse: (1) the inverse is upper triangular, and (2) the diagonal entries of the inverse are reciprocals of the diagonal entries of $A$. In other words, so the inverse has the form $$\begin{pmatrix}1 & a & b \\ 0 & 3 & c \\ 0 & 0 & 2\end{pmatrix}.$$ Then, one can simply expand the appropriate components of the equation $A A^{-1} = 0$ to solve for the three remaining unknown components $a, b, c$.