To determine the element in matrix [tex]\( M \)[/tex] at position [tex]\( M_{1,3} \)[/tex], we first need to understand the indexing of matrix elements. In general, [tex]\( M_{i,j} \)[/tex] denotes the element in the [tex]\( i \)[/tex]-th row and [tex]\( j \)[/tex]-th column of matrix [tex]\( M \)[/tex]. Rows and columns are typically numbered starting from 1.
The given matrix [tex]\( M \)[/tex] is:
[tex]\[
\left[\begin{array}{cccc}
43 & -30 & 0 & 44 \\
4 & -36 & 12 & 10 \\
16 & 31 & -17 & 39
\end{array}\right]
\][/tex]
Here, the matrix has 3 rows and 4 columns.
The notation [tex]\( M_{1,3} \)[/tex] refers to the element in the first row and the third column. To locate this element, we look at the first row, which is:
[tex]\[ \left[ 43, -30, 0, 44 \right] \][/tex]
Now, move to the third entry in this row, which is [tex]\( 0 \)[/tex].
Thus, the element [tex]\( M_{1,3} \)[/tex] is [tex]\( 0 \)[/tex].
The answer is: [tex]\( \boxed{0} \)[/tex]
