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MTH108 Business Math I

This lecture provides a review of matrix algebra, focusing on determinants, matrix operations (addition, scalar multiplication, multiplication), properties of matrix operations, matrix equations, and Cramer's rule for solving systems of linear equations.

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MTH108 Business Math I

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  1. MTH108 Business Math I Lecture 21

  2. Chapter 9 Matrix Algebra

  3. Review • Matrix; • Need, Basic facts • Types of matrices • Transpose • Matrix operations: addition; scalar multiplication; multiplication • Properties of matrix operations • Matrix equation • Representation of system of equations

  4. Today’s Topics • Determinants • Methods • Properties • Cramer’s rule

  5. Determinant Given a square matrix A, the determinant function associates with A exactly one real-valued function called the determinant, denoted by |A| or also written as Determinants are used in solving system of equations.

  6. Matrix of order 1 If A = is a matrix of order 1, then |A| = e.g. Matrix of order 2 Given a (2 × 2) matrix having the form then, |A| = e.g.

  7. This concept of finding the determinant of a (2 × 2) matrix is used in determining the determinants of matrices of orders n >2. Matrix of order 3 Given a square matrix A of order n (n > 2), to find the determinant, with a given entry of A, we associate the square matrix of order (n – 1) obtained by neglecting the row and the column in which the entry lies. E.g. if

  8. With entry , we delete the row and column containing it, i.e. the 2nd row and 1st column and we are left with an (n-1) matrix With respect to the entry we obtain an (n-1) matrix as: With respect to the entry we obtain an (n-1) matrix as:

  9. With each entry associate the number determined by the subscript of the entry: e.g. Summarizing above, 1) the determinant of the submatrices of order (n-1) are called minors and the 2) the product of the minor and the number associated with each entry is called the cofactor of the entry. e.g. the minor of and cofactor is

  10. The method of cofactors To find the determinant of any square matrix A of order n (n >2), • Select any row (or column) of the matrix. • Multiply each element in the row (or column) by its corresponding cofactor • Sum all the products (of the element with its cofactor).

  11. Here selection of row or column is very important. If we select 2nd column, then our work will be less to find the determinant of B.

  12. Properties of determinants • If all the entries in any row or column of a matrix A are zero, then |A| = 0. • If any two rows or columns of A are identical, then |A| = 0.

  13. If any two rows or columns are interchanged, the sign of the determinant changes. • If any row or column of a matrix A is multiplied by a constant k, then the determinant will be k |A|.

  14. If any multiple of one row or column is added to another row or column, the value of the determinant is unchanged. • If any row or column is a multiple of another row or column, then the determinant is zero.

  15. Given two matrices A and B, then |AB| = |A| |B|.

  16. Cramer’s Rule Given a system of linear equations, we can formulate the matrix equation AX= B where A is an n× n matrix of coefficients, X is the matrix of unknowns and B is the matrix of constants. Cramer’s rule provides a method of solving the system of n linear equations in n unknowns by using determinants.

  17. Let denote the determinant of A. To find the unknown , replace the ith column of A by B. Calculate the ratio of the determinants Where If = 0, then method is not applicable. The system may have either no solution or infinitely many solutions. If , the system has a unique solution

  18. Summary • Determinant • Determinant of matrices of different orders • The method of cofactors • Properties of determinants • Cramer’s rule Section 9.4 Follow up exercises

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