Matrices: Types and Definitions

• A matrix is a rectangular array of numbers arranged in rows and columns.
• The order of a matrix is defined by its number of rows and columns.
• An $m \times n$ matrix has $m$ rows and $n$ columns.
• $a_{ij}$ is the element in the $i^{th}$ row and $j^{th}$ column.
• Example: $\begin{bmatrix} 1 & 2 & 3 \ 4 & 5 & 6 \end{bmatrix}$ is a $2 \times 3$ matrix.

Square Matrix

• The number of rows equals the number of columns in a square matrix.
• An $n \times n$ matrix is a square matrix of order $n$.
• Example: $\begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}$ is a square matrix of order 2.

Identity Matrix

• A square matrix with 1s on the principal diagonal and 0s elsewhere is an identity matrix, denoted by $I$.
• $I_n$ represents the identity matrix of order $n$.
• Example: $I_3 = \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix}$

Null Matrix

• All elements are zero in a null matrix.
• Example: $\begin{bmatrix} 0 & 0 \ 0 & 0 \end{bmatrix}$ is a null matrix of order 2.

Diagonal Matrix

• A square matrix with non-zero elements only on the principal diagonal is a diagonal matrix.
• Example: $\begin{bmatrix} 1 & 0 & 0 \ 0 & 2 & 0 \ 0 & 0 & 3 \end{bmatrix}$ is a diagonal matrix of order 3.

Transpose of a Matrix

• The transpose of a matrix $A$ (denoted $A'$ or $A^T$) is obtained by interchanging its rows and columns.
• If $A$ is of order $m \times n$, then $A'$ is of order $n \times m$.
• Example: If $A = \begin{bmatrix} 1 & 2 & 3 \ 4 & 5 & 6 \end{bmatrix}$, then $A' = \begin{bmatrix} 1 & 4 \ 2 & 5 \ 3 & 6 \end{bmatrix}$.

Addition of Matrices

• Matrices $A$ and $B$ can be added only if they have the same order.
• If $A = [a_{ij}]{m \times n}$ and $B = [b{ij}]{m \times n}$, then $A + B = [c{ij}]{m \times n}$, where $c{ij} = a_{ij} + b_{ij}$ for all $i$ and $j$.
• Example: If $A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}$ and $B = \begin{bmatrix} 5 & 6 \ 7 & 8 \end{bmatrix}$, then $A + B = \begin{bmatrix} 6 & 8 \ 10 & 12 \end{bmatrix}$.

Subtraction of Matrices

• Matrices $A$ and $B$ can be subtracted only if they have the same order.
• If $A = [a_{ij}]{m \times n}$ and $B = [b{ij}]{m \times n}$, then $A - B = [c{ij}]{m \times n}$, where $c{ij} = a_{ij} - b_{ij}$ for all $i$ and $j$.
• Example: If $A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}$ and $B = \begin{bmatrix} 5 & 6 \ 7 & 8 \end{bmatrix}$, then $A - B = \begin{bmatrix} -4 & -4 \ -4 & -4 \end{bmatrix}$.

Multiplication of a Matrix by a Scalar

• If $A$ is a matrix and $k$ is a scalar, $kA$ is the matrix obtained by multiplying each element of $A$ by $k$.
• If $A = [a_{ij}]{m \times n}$, then $kA = [ka{ij}]_{m \times n}$.
• Example: If $A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}$ and $k = 2$, then $kA = \begin{bmatrix} 2 & 4 \ 6 & 8 \end{bmatrix}$.

Multiplication of Two Matrices

• Matrices $A$ and $B$ can be multiplied only if the number of columns of $A$ equals the number of rows of $B$.
• If $A = [a_{ij}]{m \times n}$ and $B = [b{ij}]{n \times p}$, then $AB = [c{ij}]{m \times p}$, where $c{ij} = \sum_{k=1}^{n} a_{ik}b_{kj}$ for all $i$ and $j$.
• Example: If $A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}$ and $B = \begin{bmatrix} 5 & 6 \ 7 & 8 \end{bmatrix}$, then $AB = \begin{bmatrix} 19 & 22 \ 43 & 50 \end{bmatrix}$.

Explore matrices, their order, and element notation. Learn about square matrices where rows equal columns. Discover identity matrices with 1s on the diagonal and null matrices with all zero elements.