Answer :

Sure, let's work through the given problem step-by-step.

We are given the matrix [tex]\( B \)[/tex]:

[tex]\[ B = \begin{bmatrix} -1 & 2 & 7 \\ -5 & 1 & 0 \\ 2 & 3 & 2 \end{bmatrix} \][/tex]

Here is a detailed step-by-step construction of the matrix [tex]\( B \)[/tex]:

1. Identify the components of the matrix: The matrix [tex]\( B \)[/tex] is a 3x3 matrix, meaning it has 3 rows and 3 columns.

2. Fill in each element of the matrix:

- The element in the first row and first column (1,1) is [tex]\(-1\)[/tex].
- The element in the first row and second column (1,2) is [tex]\(2\)[/tex].
- The element in the first row and third column (1,3) is [tex]\(7\)[/tex].

This fills the first row: [tex]\([-1, 2, 7]\)[/tex].

- The element in the second row and first column (2,1) is [tex]\(-5\)[/tex].
- The element in the second row and second column (2,2) is [tex]\(1\)[/tex].
- The element in the second row and third column (2,3) is [tex]\(0\)[/tex].

This fills the second row: [tex]\([-5, 1, 0]\)[/tex].

- The element in the third row and first column (3,1) is [tex]\(2\)[/tex].
- The element in the third row and second column (3,2) is [tex]\(3\)[/tex].
- The element in the third row and third column (3,3) is [tex]\(2\)[/tex].

This fills the third row: [tex]\([2, 3, 2]\)[/tex].

3. Combine all rows to form the matrix [tex]\( B \)[/tex]:

[tex]\[ B = \begin{bmatrix} -1 & 2 & 7 \\ -5 & 1 & 0 \\ 2 & 3 & 2 \end{bmatrix} \][/tex]

Thus, the matrix [tex]\( B \)[/tex] is:

[tex]\[ B = \begin{bmatrix} -1 & 2 & 7 \\ -5 & 1 & 0 \\ 2 & 3 & 2 \end{bmatrix} \][/tex]

And there is our detailed construction of the given matrix!