To determine the type of matrix Mateus has created, let’s follow these steps:
### Step 1: Construct the Matrix
Based on the given elements, we can construct the matrix as:
[tex]\[
A = \begin{pmatrix}
32 & 10 & -9 \\
5 & 7.5 & 4 \\
-0.5 & 6 & 16
\end{pmatrix}
\][/tex]
### Step 2: Check if the Matrix is Square
A matrix is square if it has the same number of rows and columns. Matrix [tex]\( A \)[/tex] has dimensions [tex]\( 3 \times 3 \)[/tex], because it has 3 rows and 3 columns.
Since the number of rows equals the number of columns, matrix [tex]\( A \)[/tex] is a square matrix.
### Step 3: Check if the Matrix is Diagonal
A diagonal matrix is a special kind of square matrix where all elements off the main diagonal are zero. The main diagonal elements of matrix [tex]\( A \)[/tex] are 32, 7.5, and 16.
To be a diagonal matrix, all other elements (non-diagonal elements) in [tex]\( A \)[/tex] should be zero.
Let’s look at the non-diagonal elements of matrix [tex]\( A \)[/tex]:
- [tex]\( a_{12} = 10 \)[/tex]
- [tex]\( a_{13} = -9 \)[/tex]
- [tex]\( a_{21} = 5 \)[/tex]
- [tex]\( a_{23} = 4 \)[/tex]
- [tex]\( a_{31} = -0.5 \)[/tex]
- [tex]\( a_{32} = 6 \)[/tex]
Since these elements are not zero, matrix [tex]\( A \)[/tex] is not a diagonal matrix.
### Conclusion
Based on our analysis:
- Matrix [tex]\( A \)[/tex] is a square matrix.
- Matrix [tex]\( A \)[/tex] is not a diagonal matrix.
So, the answer is that Mateus created a square matrix.
