1. What Is a Square Matrix?
A square matrix is a matrix in which the number of rows is equal to the number of columns. Its order is written as n × n.
Examples include matrices of order 1 × 1, 2 × 2, 3 × 3, 4 × 4, etc. These are called square because they form a perfect square shape when drawn.
In general, a square matrix looks like:
\( A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} \end{bmatrix} \)
2. Understanding the Structure of a Square Matrix
Every square matrix has a special diagonal called the main diagonal or principal diagonal. It consists of the elements:
\( a_{11}, a_{22}, a_{33}, \dots, a_{nn} \)
Square matrices allow us to define important concepts like determinants, inverses, eigenvalues, and many other advanced operations.
2.1. Example of a Square Matrix
Here is a 3 × 3 square matrix:
\( B = \begin{bmatrix} 2 & -1 & 4 \\ 0 & 3 & 7 \\ 5 & 6 & 1 \end{bmatrix} \)
It has:
- 3 rows
- 3 columns
Since both are equal, it is a square matrix.
2.2. More Examples
- \( [5] \) — a 1 × 1 square matrix.
- \( \begin{bmatrix} 1 & 0 \\ -2 & 3 \end{bmatrix} \) — a 2 × 2 square matrix.
- \( \begin{bmatrix} 4 & 7 & 1 \\ 9 & -3 & 2 \\ 6 & 0 & 5 \end{bmatrix} \) — a 3 × 3 square matrix.
3. Special Features of Square Matrices
Square matrices are extremely important in mathematics because they allow several operations that are not possible with non-square matrices. Some key features are:
- They have a determinant.
- They may have an inverse (if determinant ≠ 0).
- They can represent linear transformations.
- They form the basis for defining symmetric, skew-symmetric, diagonal, and identity matrices.
4. Main Diagonal and Its Importance
The main diagonal plays a major role in many definitions. For example:
- If all off-diagonal elements are zero, it becomes a diagonal matrix.
- If diagonal elements are all equal, it becomes a scalar matrix.
- If diagonal elements are all 1, it becomes an identity matrix.
- If entries mirror across the diagonal, you get symmetric matrices.
So the structure of square matrices opens the door to several advanced types of matrices.