Question

If matrix $A = \left( {\begin{array}{*{20}{c}}
  1&0&{ - 1} \\
  3&4&5 \\
  0&6&7
\end{array}} \right)$ and its inverse is denoted by ${A^1} = \left( {\begin{array}{*{20}{c}}
  {{a_{11}}}&{{a_{12}}}&{{a_{13}}} \\
  {{a_{21}}}&{{a_{22}}}&{{a_{23}}} \\
  {{a_{31}}}&{{a_{32}}}&{{a_{33}}}
\end{array}} \right)$ , then the value of ${A_{23}} = $

Answer 1

Hint: Here we will use some of the matrix rules and matrix formulas. Then we will substitute the values in the matrix. These values are denoted by the adjacent. Then that will be concluded using some rules. Finally, we will get the answer.

Complete step-by-step answer:
The question given,
$A = \left( {\begin{array}{*{20}{c}}
  1&0&{ - 1} \\
  3&4&5 \\
  0&6&7
\end{array}} \right)$
${A^{ - 1}} = \dfrac{{adjA}}{{\det A}}$ (here $adjA$ means Adjacent $A$ and det $A$ means determinant $A$ )
Here $\det A = 1(28 - 30) - 1(18)$(formula of determinant a $A = \left( {\begin{array}{*{20}{c}}
a&b&c \\
d&e&f \\
g&h&i
\end{array}} \right)$ is \[\left| A \right|{\text{ }} = {\text{ }}a\left( {ei{\text{ }} - {\text{ }}fh} \right){\text{ }} - {\text{ }}b\left( {di{\text{ }} - {\text{ }}fg} \right){\text{ }} + {\text{ }}c\left( {dh{\text{ }} - {\text{ }}eg} \right)\] )
Using above formula, we will get the $\det A$
$
= - 2 - 18 \\
= - 20 \\
 $
Then we will solve the $adjA$.
Here we will find adjacent of matrix
We have a method for finding adjacent matrices. Using that method find the adjacent matrix.
$adjA = \left( {\begin{array}{*{20}{c}}
{ - 2}&{ - 6}&4 \\
{ - 21}&7&{ - 8} \\
  {18}&{ - 6}&4
\end{array}} \right)$
\[{A^{ - 1}} = \dfrac{{ - 1}}{{20}}\left( {\begin{array}{*{20}{c}}
{ - 2}&{ - 6}&4 \\
{ - 21}&7&{ - 8} \\
  {18}&{ - 6}&4
\end{array}} \right)\]
We will solve the above adjacent matrix we will get
${a_{23}} = \dfrac{{ - 8}}{{ - 20}} = \dfrac{2}{5}$

Here the answer will be $\dfrac{2}{5}$

Additional information:
An adjacency matrix is a square matrix used to represent a finite graph. The elements of the matrix indicate whether pairs of vertices are adjacent or not in the graph. In the special case of a finite simple graph, the adjacency matrix is a \[\left( {0,1} \right)\] . Adjacency matrix is a means of representing which vertices of a graph are adjacent to which other vertices. Another matrix representation for a graph is the incidence matrix.

Note: Two vertices are said to be adjacent or neighbor if it supports at least one common edge. To fill the adjacency matrix, we look at the name of the vertex in row and column. If those vertices are connected by an edge or more, we count the number of edges and put this number as a matrix element. We will find the answer using these types of formula.