2.1 Matrix Order and Types

A matrix is a rectangular array of numbers (real or complex) arranged in rows and columns, enclosed in brackets.

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Order of a Matrix

The order (or size) of a matrix is written as $m\times n$, where:

• $m$ = number of rows
• $n$ = number of columns

Example: A matrix with 3 rows and 4 columns has order $3\times 4$.

$A={\left[\begin{array}{cccc}1& 2& 3& 4\\ 5& 6& 7& 8\\ 9& 10& 11& 12\end{array}\right]}_{3\times 4}$

The element in row $i$ and column $j$ is denoted $a_{ij}$.

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Types of Matrices

1. Row Matrix

A matrix with exactly one row: order $1\times n$. $R={\left[\begin{array}{ccc}3& -1& 7\end{array}\right]}_{1\times 3}$

2. Column Matrix

A matrix with exactly one column: order $m\times 1$. $C={\left[\begin{array}{c}2\\ -5\\ 0\end{array}\right]}_{3\times 1}$

3. Square Matrix

A matrix where the number of rows equals the number of columns: $m=n$. $S={\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]}_{2\times 2}$

4. Rectangular Matrix

A matrix where $m\ne n$.

5. Zero (Null) Matrix

All entries are zero, denoted $O$. $O=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$

6. Diagonal Matrix

A square matrix where all entries off the main diagonal are zero. $D=\left[\begin{array}{ccc}3& 0& 0\\ 0& -1& 0\\ 0& 0& 5\end{array}\right]$

7. Scalar Matrix

A diagonal matrix where all main diagonal entries are equal to the same constant $k$. $S=\left[\begin{array}{ccc}k& 0& 0\\ 0& k& 0\\ 0& 0& k\end{array}\right]$

8. Identity (Unit) Matrix

A scalar matrix where $k=1$, denoted $I$. $I_3=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$

9. Upper Triangular Matrix

All entries below the main diagonal are zero ($a_{ij}=0$ for $i>j$). $U=\left[\begin{array}{ccc}1& 2& 3\\ 0& 4& 5\\ 0& 0& 6\end{array}\right]$

10. Lower Triangular Matrix

All entries above the main diagonal are zero ($a_{ij}=0$ for $i<j$). $L=\left[\begin{array}{ccc}1& 0& 0\\ 2& 3& 0\\ 4& 5& 6\end{array}\right]$

11. Transpose of a Matrix

The transpose of $A$ (denoted $A^T$) is obtained by interchanging rows and columns: $(A^T{)}_{ij}=a_{ji}$.

$A=\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\end{array}\right]⟹A^T=\left[\begin{array}{cc}1& 4\\ 2& 5\\ 3& 6\end{array}\right]$

12. Symmetric Matrix

A square matrix where $A^T=A$ (i.e., $a_{ij}=a_{ji}$ for all $i,j$). $A=\left[\begin{array}{ccc}1& 2& 3\\ 2& 5& 4\\ 3& 4& 6\end{array}\right]$

13. Skew-Symmetric Matrix

A square matrix where $A^T=-A$ (i.e., $a_{ij}=-a_{ji}$).

Key property: All diagonal entries must be zero, since $a_{ii}=-a_{ii}\Rightarrow a_{ii}=0$.

$A=\left[\begin{array}{ccc}0& -2& 3\\ 2& 0& -1\\ -3& 1& 0\end{array}\right]$

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Summary Table

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