Certainly! Let's explore and understand the matrix [tex]\( A \)[/tex] given by:
[tex]\[
A = \begin{pmatrix}
3 & 1 \\
-3 & -1
\end{pmatrix}
\][/tex]
To provide a detailed solution, we'll analyze the structure of this matrix step-by-step:
1. Matrix Definition:
- Matrix Size: The matrix [tex]\( A \)[/tex] is a 2x2 matrix, which means it has 2 rows and 2 columns.
- Elements of the Matrix: Each element in the matrix is denoted by [tex]\( a_{ij} \)[/tex] where [tex]\( i \)[/tex] represents the row number and [tex]\( j \)[/tex] represents the column number.
In the matrix [tex]\( A \)[/tex]:
- [tex]\( a_{11} = 3 \)[/tex]
- [tex]\( a_{12} = 1 \)[/tex]
- [tex]\( a_{21} = -3 \)[/tex]
- [tex]\( a_{22} = -1 \)[/tex]
2. Matrix Representation:
Let's explicitly list the elements of the matrix:
- The first row consists of the elements 3 and 1.
- The second row consists of the elements -3 and -1.
Therefore, the matrix [tex]\( A \)[/tex] is represented as:
[tex]\[
A = \begin{pmatrix}
3 & 1 \\
-3 & -1
\end{pmatrix}
\][/tex]
3. Understanding the Matrix:
- Each element has a specific position in the matrix, which affects various linear transformations when this matrix is applied to a vector.
### Conclusion
We have expressed and verified the matrix [tex]\( A \)[/tex] given by:
[tex]\[
A = \begin{pmatrix}
3 & 1 \\
-3 & -1
\end{pmatrix}
\][/tex]
This concludes our detailed step-by-step exploration of the structure and elements of matrix [tex]\( A \)[/tex].
