Given a matrix [tex]\( A \)[/tex]:
[tex]\[ A = \begin{bmatrix}
3 & 1 & -2 \\
4 & 2 & 5 \\
-6 & 3 & -1
\end{bmatrix} \][/tex]
Let's analyze the structure and properties of this matrix step by step.
### Structure of the Matrix:
1. Dimensions: Matrix [tex]\( A \)[/tex] is a [tex]\( 3 \times 3 \)[/tex] matrix, meaning it has 3 rows and 3 columns.
2. Elements: The elements of the matrix are given as follows:
[tex]\[ a_{11} = 3, \; a_{12} = 1, \; a_{13} = -2 \][/tex]
[tex]\[ a_{21} = 4, \; a_{22} = 2, \; a_{23} = 5 \][/tex]
[tex]\[ a_{31} = -6, \; a_{32} = 3, \; a_{33} = -1 \][/tex]
### Elements of Each Row and Column:
- First Row Elements:
[tex]\[ \text{Row 1: } 3, 1, -2 \][/tex]
- Second Row Elements:
[tex]\[ \text{Row 2: } 4, 2, 5 \][/tex]
- Third Row Elements:
[tex]\[ \text{Row 3: } -6, 3, -1 \][/tex]
- First Column Elements:
[tex]\[ \text{Column 1: } 3, 4, -6 \][/tex]
- Second Column Elements:
[tex]\[ \text{Column 2: } 1, 2, 3 \][/tex]
- Third Column Elements:
[tex]\[ \text{Column 3: } -2, 5, -1 \][/tex]
### Matrix Representation:
The matrix can be written element-wise in a tabular form as:
[tex]\[ A = \begin{bmatrix}
3 & 1 & -2 \\
4 & 2 & 5 \\
-6 & 3 & -1
\end{bmatrix} \][/tex]
### Matrix Properties:
- Square Matrix: Since the number of rows is equal to the number of columns ([tex]\(3 \times 3\)[/tex]), [tex]\(A\)[/tex] is a square matrix.
- Determinant: To find the determinant, one typically uses the formula specific to [tex]\(3 \times 3\)[/tex] matrices:
[tex]\[
\text{det}(A) = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31})
\][/tex]
(Calculations can be carried out as another step if needed.)
- Inverse and Eigenvalues: Likewise, the inverse matrix and eigenvalues can be computed using standard linear algebra techniques.
We have successfully analyzed and represented the given matrix [tex]\( A \)[/tex] in a detailed manner.
