Mathematics for Business I: Matrices and Determinants

Introduction

• Professor: Sarai Vera Rodríguez

• UCAM Universid Católica San Antonio

Concept of Matrix

• A matrix is a set of elements arranged in rows and columns.

• Matrices used in this course will only include real numbers.

• The order of a matrix is defined by the number of its rows and columns:

  • A matrix with m rows and n columns has an order of m x n.

Matrix Representation

• Matrix elements are represented as aij, where i indicates the row and j indicates the column.

• Matrix A can be expressed as:

  A = \begin{pmatrix} a_{11} & a_{12} & a_{13} & \ldots & a_{1n} \\ a_{21} & a_{22} & a_{23} & \ldots & a_{2n} \\ a_{31} & a_{32} & a_{33} & \ldots & a_{3n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & a_{m3} & \ldots & a_{mn} \end{pmatrix}

  • The matrix is expressed as amxn.

Matrix as a Set of Vectors

• A matrix can also be viewed as a set of vectors:

  • A matrix comprises m row vectors and n column vectors.

  • The notation for expressing rows and columns is:

    egin{pmatrix} a_{11} & a_{12} & a_{13} & \ldots \end{pmatrix} (row)

    • or egin{pmatrix} a_{1j} \ a_{2j} \ a_{3j} \ \vdots \end{pmatrix} (column)

Matrix Equality

• Two matrices A and B are equal if:

  1. They have the same order.

  2. Corresponding elements are equal: aij = bij, for all i,j.

Types of Matrices

• A rectangular matrix is when the number of rows does not equal the number of columns.

  • Example: A matrix with 3 rows and 5 columns.

• A row matrix has only one row.

• A column matrix has only one column.

• A square matrix has the same number of rows and columns (order n).

  • Example: For a square matrix of order 4:

  • A = \begin{pmatrix} 3 & 8 & 3 & 4 \ 0 & 1 & 2 & 3 \ 2 & 0 & 2 & 5 \ 1 & 7 & 2 & 6 \end{pmatrix}

• The main diagonal consists of elements aii, e.g., 1, 0, 2, 4 in a square matrix.

Triangular Matrices

• A superior triangular matrix has zero elements below the main diagonal (aij=0 for i > j).

• An inferior triangular matrix has zeros above the main diagonal (aij=0 for i < j).

  • Example of each type illustrated.

• A diagonal matrix is both superior and inferior triangular; all non-diagonal elements are 0.

  • A scalar matrix has all diagonal elements equal to the same number.

  • A unit matrix (identity matrix) has diagonal elements equal to 1.

Operations on Matrices

Sum of Matrices

• The sum of two matrices A and B of the same order:

  • Resulting matrix has elements (i,j) equal to the sum of A's and B's elements:

  • A + B = (a_{ij} + b_{ij})

  • Associativity, identity and inverse properties.

Scalar Multiplication

• The scalar multiplication k·A scales elements of matrix A by k:

  k·A = (k·a_{ij})

  • Examples illustrating scalar multiplication properties.

Matrix Multiplication

• A matrix A of order mxn and B of order nxp can be multiplied:

  • The resultant matrix C = AB is of order mxp.

  • Each element c_{ij} in C is calculated as the dot product of the i-th row of A and the j-th column of B.

• Properties of matrix multiplication include left and right distributivity, associativity.

Transpose Matrix

• The transpose of matrix A switches rows with columns:

  • Denoted as A’ or At.

  • Properties of transpose matrices include: (A+B)t = At + Bt, (AB)t = BtAt, and (At)t = A.

Square Matrices

• A square matrix A is:

  1. symmetric if it equals its transpose (A = At).

  2. asymmetric if A = -At.

  3. idempotent if A^2 = A.

Determinant

• The determinant of a square matrix represents scalar properties of the matrix.

• Rules and properties for calculating determinants include:

  • Determinant equals zero if rows/columns are linearly dependent.

  • Determinant of triangular matrices equals the product of diagonal elements.

Inverse Matrix

• A square matrix A has an inverse matrix X if AX = XA = I, where I is the identity matrix.

• A matrix is regular (invertible) if its determinant is non-zero.

Orthogonal Matrices

• A square matrix A is orthogonal if AAt = I; all row vectors are orthonormal.

Matrix Rank

• The rank of a matrix A is the maximum number of linearly independent row or column vectors.

• It is calculated based on finding non-null minors.