Certainly! Let's analyze and identify the entries of matrix [tex]\( B \)[/tex] such that they correspond to the given notations:
We start with the matrix [tex]\( B \)[/tex] given below:
[tex]\[
B = \begin{pmatrix}
4 & -2 & 0 & 2 \\
-1 & 6 & 1 & -3 \\
5 & 7 & 3 & 8
\end{pmatrix}
\][/tex]
### Entry [tex]\( B_{13} \)[/tex]
The notation [tex]\( B_{ij} \)[/tex] represents the element in the [tex]\(i\)[/tex]-th row and [tex]\(j\)[/tex]-th column of the matrix. Here, [tex]\( B_{13} \)[/tex]:
- [tex]\(i = 1\)[/tex] (1st row)
- [tex]\(j = 3\)[/tex] (3rd column)
So, [tex]\( B_{13} \)[/tex] is the element at the first row and third column:
[tex]\[
B_{13} = 0
\][/tex]
### Entry [tex]\( B_{32} \)[/tex]
Considering [tex]\( B_{32} \)[/tex]:
- [tex]\(i = 3\)[/tex] (3rd row)
- [tex]\(j = 2\)[/tex] (2nd column)
So, [tex]\( B_{32} \)[/tex] is the element at the third row and second column:
[tex]\[
B_{32} = 7
\][/tex]
### Entry [tex]\( B_7 \)[/tex]
We need to identify where 7 is located in the matrix. Looking at matrix [tex]\( B \)[/tex]:
[tex]\[
\begin{pmatrix}
4 & -2 & 0 & 2 \\
-1 & 6 & 1 & -3 \\
5 & 7 & 3 & 8
\end{pmatrix}
\][/tex]
We observe that the entry 7 is located at the third row and second column. Thus,
[tex]\[
B_7 = B_{32}
\][/tex]
### Entry [tex]\( B_{23} \)[/tex]
Considering [tex]\( B_{23} \)[/tex]:
- [tex]\(i = 2\)[/tex] (2nd row)
- [tex]\(j = 3\)[/tex] (3rd column)
So, [tex]\( B_{23} \)[/tex] is the element at the second row and third column:
[tex]\[
B_{23} = 1
\][/tex]
To summarize:
- [tex]\( B_{13} \)[/tex]: [tex]\( 0 \)[/tex]
- [tex]\( B_{32} \)[/tex]: [tex]\( 7 \)[/tex]
- [tex]\( B_7 \)[/tex]: [tex]\( 7 \)[/tex] (matches entry [tex]\( B_{32} \)[/tex])
- [tex]\( B_{23} \)[/tex]: [tex]\( 1 \)[/tex]
Thus, we determine that 7 is the entry at [tex]\( B_{32} \)[/tex].
