Matrices Definitions and Types

Matrices Definition

• A matrix is a rectangular array of numbers in rows and columns.
• The order of the matrix is the number of rows and columns.
• An $m \times n$ matrix consists of m rows and n columns.
• Matrices are denoted by uppercase letters.
• Matrix elements are denoted by lowercase letters with subscripts for row and column.
• $A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \ a_{21} & a_{22} & \cdots & a_{2n} \ \vdots & \vdots & \ddots & \vdots \ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix}$
• $a_{ij}$ is the element in the $i$-th row and $j$-th column.

Types of Matrices

• Row Matrix: A matrix with only one row. Example: $A = \begin{bmatrix} 1 & 5 & 9 \end{bmatrix}_{1 \times 3}$
• Column Matrix: A matrix with only one column. Example: $A = \begin{bmatrix} 1 \ 6 \ 9 \end{bmatrix}_{3 \times 1}$
• Zero or Null Matrix: All elements are zero. Example: $A = \begin{bmatrix} 0 & 0 \ 0 & 0 \end{bmatrix}_{2 \times 2}$
• Square Matrix: Number of rows equals the number of columns. Example: $A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}_{2 \times 2}$
• Diagonal Matrix: Square matrix with all non-diagonal elements zero. Example: $A = \begin{bmatrix} 3 & 0 & 0 \ 0 & 5 & 0 \ 0 & 0 & 8 \end{bmatrix}_{3 \times 3}$
• Scalar Matrix: Diagonal matrix with equal diagonal elements. Example: $A = \begin{bmatrix} 3 & 0 & 0 \ 0 & 3 & 0 \ 0 & 0 & 3 \end{bmatrix}_{3 \times 3}$
• Unit or Identity Matrix: Diagonal matrix with diagonal elements equal to one, denoted by $I$. Example: $I = \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix}_{3 \times 3}$
• Triangular Matrix:
  • Upper Triangular Matrix: Square matrix with all elements below the diagonal zero. Example: $A = \begin{bmatrix} 1 & 2 & 3 \ 0 & 4 & 5 \ 0 & 0 & 6 \end{bmatrix}_{3 \times 3}$
  • Lower Triangular Matrix: Square matrix with all elements above the diagonal zero. Example: $A = \begin{bmatrix} 1 & 0 & 0 \ 2 & 3 & 0 \ 4 & 5 & 6 \end{bmatrix}_{3 \times 3}$

Transpose of a Matrix

• The transpose of matrix $A$, denoted as $A'$ or $A^T$, is obtained by interchanging rows and columns.
• If $A = \begin{bmatrix} 1 & 2 & 3 \ 4 & 5 & 6 \end{bmatrix}$, then $A^T = \begin{bmatrix} 1 & 4 \ 2 & 5 \ 3 & 6 \end{bmatrix}$

Properties of Transpose

• $(A^T)^T = A$
• $(A + B)^T = A^T + B^T$
• $(kA)^T = kA^T$, where $k$ is a scalar.
• $(AB)^T = B^T A^T$

Symmetric Matrix

• A square matrix $A$ is symmetric if $A^T = A$. That is, $a_{ij} = a_{ji}$ for all $i$ and $j$.
• Example: $A = \begin{bmatrix} 1 & 2 & 3 \ 2 & 4 & 5 \ 3 & 5 & 6 \end{bmatrix}$

Skew-Symmetric Matrix

• A square matrix $A$ is skew-symmetric if $A^T = -A$. That is, $a_{ij} = -a_{ji}$ for all $i$ and $j$.
• The diagonal elements of a skew-symmetric matrix are always zero.
• Example: $A = \begin{bmatrix} 0 & 2 & -3 \ -2 & 0 & -5 \ 3 & 5 & 0 \end{bmatrix}$

Matrix Algebra

Addition of Matrices

• Two matrices $A$ and $B$ are conformable for addition if they have the same order.
• If $A = [a_{ij}]{m \times n}$ and $B = [b{ij}]{m \times n}$, then $A + B = [a{ij} + b_{ij}]_{m \times n}$.

Properties of Matrix Addition

• Commutative: $A + B = B + A$
• Associative: $(A + B) + C = A + (B + C)$
• Additive Identity: $A + O = A$, where $O$ is the null matrix.
• Additive Inverse: $A + (-A) = O$

Subtraction of Matrices

• If $A = [a_{ij}]{m \times n}$ and $B = [b{ij}]{m \times n}$, then $A - B = [a{ij} - b_{ij}]_{m \times n}$.

Multiplication of a Matrix by a Scalar

• If $A$ is a matrix and $k$ is a scalar, then $kA = [ka_{ij}]_{m \times n}$.

Properties of Scalar Multiplication

• $k(A + B) = kA + kB$
• $(k + l)A = kA + lA$
• $(kl)A = k(lA) = l(kA)$
• $1 \cdot A = A$

Multiplication of Matrices

• Two matrices $A$ and $B$ are conformable for multiplication if the number of columns in $A$ equals the number of rows in $B$.
• If $A = [a_{ij}]{m \times n}$ and $B = [b{jk}]{n \times p}$, then $AB = C = [c{ik}]{m \times p}$, where $c{ik} = \sum_{j=1}^{n} a_{ij} b_{jk}$.

Properties of Matrix Multiplication

• Not commutative in general: $AB \neq BA$
• Associative: $(AB)C = A(BC)$
• Distributive over addition: $A(B + C) = AB + AC$ and $(A + B)C = AC + BC$
• Multiplicative Identity: $AI = IA = A$, where $I$ is the identity matrix.

Determinant of a Matrix

• Determinant of a $2 \times 2$ Matrix: If $A = \begin{bmatrix} a & b \ c & d \end{bmatrix}$, then $|A| = ad - bc$.
• Determinant of a $3 \times 3$ Matrix: If $A = \begin{bmatrix} a_{11} & a_{12} & a_{13} \ a_{21} & a_{22} & a_{23} \ a_{31} & a_{32} & a_{33} \end{bmatrix}$, then $|A| = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31})$

Properties of Determinants

• Zero Determinant: If all elements of a row or column are zero, the determinant is zero.
• Row or Column Interchange: Changes the sign of the determinant.
• Identical Rows or Columns: The determinant is zero.
• Scalar Multiplication of Row or Column: The determinant is multiplied by $k$.
• Triangular Matrix: The determinant is the product of the diagonal elements.

Adjoint of a Matrix

• The adjoint of a square matrix $A$ is the transpose of the cofactor matrix of $A$.
• If $A = [a_{ij}]$, then $\text{adj}(A) = [C_{ij}]^T$, where $C_{ij}$ is the cofactor of $a_{ij}$.

Properties of Adjoint

• $A(\text{adj }A) = (\text{adj }A)A = |A|I$
• $\text{adj}(AB) = (\text{adj }B)(\text{adj }A)$
• $|\text{adj }A| = |A|^{n-1}$, where $n$ is the order of the matrix $A$.

Inverse of a Matrix

• A square matrix $A$ is invertible (or non-singular) if there exists a matrix $B$ such that $AB = BA = I$. $B$ is the inverse of $A$, denoted $A^{-1}$.
• $A^{-1} = \frac{1}{|A|} \text{adj }A$

Properties of Inverse

• $(A^{-1})^{-1} = A$
• $(AB)^{-1} = B^{-1} A^{-1}$
• $(A^T)^{-1} = (A^{-1})^T$
• $A A^{-1} = A^{-1} A = I$