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# Matrices ## Objectives After completing this section, you should be able to - Define a matrix and identify its elements, rows, and columns. - Determine the order of a matrix. - Identify different types of matrices such as row matrix, column matrix, square matrix, diagonal matrix, scalar matrix,...

# Matrices ## Objectives After completing this section, you should be able to - Define a matrix and identify its elements, rows, and columns. - Determine the order of a matrix. - Identify different types of matrices such as row matrix, column matrix, square matrix, diagonal matrix, scalar matrix, identity matrix, and zero matrix. - Perform basic matrix operations: addition, subtraction, and scalar multiplication. - Understand the properties of matrix addition and scalar multiplication. ## Definition of a Matrix A matrix is a rectangular array of numbers arranged in rows and columns. Each number in a matrix is called an element or entry. Matrices are typically denoted by uppercase letters such as A, B, C, etc. ### General Form of a Matrix $$ A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix} $$ Where: - \( a_{ij} \) represents the element in the \( i \)-th row and \( j \)-th column. - \( m \) is the number of rows. - \( n \) is the number of columns. ### Order of a Matrix The order of a matrix is defined by the number of rows and columns it contains. A matrix with \( m \) rows and \( n \) columns is said to be of order \( m \times n \) (read as "m by n"). ### Example: $$ A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix} $$ Matrix A has 2 rows and 3 columns, so its order is \( 2 \times 3 \). ## Types of Matrices 1. **Row Matrix:** A matrix with only one row. Example: \( A = \begin{bmatrix} 1 & 2 & 3 \end{bmatrix} \) is a row matrix of order \( 1 \times 3 \). 2. **Column Matrix:** A matrix with only one column. Example: \( A = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix} \) is a column matrix of order \( 3 \times 1 \). 3. **Square Matrix:** A matrix in which the number of rows is equal to the number of columns. Example: \( A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \) is a square matrix of order \( 2 \times 2 \). 4. **Diagonal Matrix:** A square matrix in which all the elements outside the main diagonal are zero. Example: \( A = \begin{bmatrix} 1 & 0 \\ 0 & 4 \end{bmatrix} \) is a diagonal matrix of order \( 2 \times 2 \). 5. **Scalar Matrix:** A diagonal matrix in which all the elements on the main diagonal are equal. Example: \( A = \begin{bmatrix} 5 & 0 \\ 0 & 5 \end{bmatrix} \) is a scalar matrix of order \( 2 \times 2 \). 6. **Identity Matrix:** A scalar matrix in which all the elements on the main diagonal are equal to 1. It is denoted by I. Example: \( I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \) is an identity matrix of order \( 2 \times 2 \). 7. **Zero Matrix:** A matrix in which all the elements are zero. It is denoted by O. Example: \( O = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} \) is a zero matrix of order \( 2 \times 2 \). ## Basic Matrix Operations ### Addition of Matrices Matrices can be added if they have the same order. The sum of two matrices A and B of the same order is a matrix C, where each element \( c_{ij} \) is the sum of the corresponding elements \( a_{ij} \) and \( b_{ij} \). If \( A = [a_{ij}] \) and \( B = [b_{ij}] \) are two matrices of order \( m \times n \), then \( C = A + B \) is also of order \( m \times n \) and \( c_{ij} = a_{ij} + b_{ij} \) for all \( i \) and \( j \). Example: $$ A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}, \quad B = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix} $$ $$ A + B = \begin{bmatrix} 1+5 & 2+6 \\ 3+7 & 4+8 \end{bmatrix} = \begin{bmatrix} 6 & 8 \\ 10 & 12 \end{bmatrix} $$ ### Subtraction of Matrices Similar to addition, matrices can be subtracted if they have the same order. The difference of two matrices A and B of the same order is a matrix C, where each element \( c_{ij} \) is the difference of the corresponding elements \( a_{ij} \) and \( b_{ij} \). If \( A = [a_{ij}] \) and \( B = [b_{ij}] \) are two matrices of order \( m \times n \), then \( C = A - B \) is also of order \( m \times n \) and \( c_{ij} = a_{ij} - b_{ij} \) for all \( i \) and \( j \). Example: $$ A = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}, \quad B = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} $$ $$ A - B = \begin{bmatrix} 5-1 & 6-2 \\ 7-3 & 8-4 \end{bmatrix} = \begin{bmatrix} 4 & 4 \\ 4 & 4 \end{bmatrix} $$ ### Scalar Multiplication Scalar multiplication involves multiplying a matrix by a constant (scalar). If A is a matrix and k is a scalar, then the matrix kA is obtained by multiplying every element of A by k. If \( A = [a_{ij}] \) is a matrix of order \( m \times n \), then \( kA = [ka_{ij}] \) is also of order \( m \times n \) for all \( i \) and \( j \). Example: $$ A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}, \quad k = 2 $$ $$ kA = 2 \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} = \begin{bmatrix} 2 \times 1 & 2 \times 2 \\ 2 \times 3 & 2 \times 4 \end{bmatrix} = \begin{bmatrix} 2 & 4 \\ 6 & 8 \end{bmatrix} $$ ## Properties of Matrix Addition 1. **Commutative Law:** For any two matrices A and B of the same order, \( A + B = B + A \). 2. **Associative Law:** For any three matrices A, B, and C of the same order, \( (A + B) + C = A + (B + C) \). 3. **Existence of Additive Identity:** For any matrix A, there exists a zero matrix O of the same order such that \( A + O = A = O + A \). 4. **Existence of Additive Inverse:** For any matrix A, there exists a matrix -A such that \( A + (-A) = O = (-A) + A \), where -A is the negative of matrix A. ## Properties of Scalar Multiplication For any matrices A and B of the same order, and any scalars k and l: 1. \( k(A + B) = kA + kB \) 2. \( (k + l)A = kA + lA \) 3. \( (kl)A = k(lA) = l(kA) \) 4. \( 1A = A \)