Unit 5DeterminantsIntroductionIn this unit the determinant function is studied. This function was introduced earlier for 2 x 2 squarematrices in section 1.6 to help in the calculation of the cross product of two vectors inR3.In thischapter we extend the definition of a determinant to any size square matrix and consider severaldifferent methods for calculating the value of the determinant. This is followed by some usefulapplications involving the determinant function itself such as determining whether A is invertible ornoninvertible. An explicit formula for A–1exists that involves the determinant of A. Also, somesystems of linear equations have a solution that can be expressed in terms of determinants.A note on determinants as a functionIn mathematics a function is a rule that assigns to each object from one set, a unique object in asecond set. Although many kinds of rules (or functions) can be constructed, the importance of afunction depends upon its applicability. In the case of the determinant function, the objects of thefirst set are the square matrices, and the objects in the second set are the real numbers. Becausethe determinant function has a variety of applications in various branches of mathematics, thedeterminant is considered to be an important topic, worthy of study at an early stage of a person'smathematical education.Learning objectivesUpon completion of this unit you should be able to:•define the determinant of a square matrix;•calculate the ijthcofactor of a square matrix;•evaluate the determinant using a row or column cofactor expansion;•determine the effect on the value of a determinant if an elementary row operation is performed;•use the determinant to determine if a square matrix is invertible or not invertible;•find the adjoint matrix for a given square matrix;•use the adjoint and determinant to calculate the inverse of a matrix;•apply Cramer's rule to solve a system ofnlinear equations innvariables; and•use the determinant to calculate the volume of a parallelepiped.Unit activities1.Review the fifth assignment to get an idea of the types of problems you will be required to solvefor this unit. Check the due date for the assignment in the course syllabus.2.Read each section in unit 5 and carefully work through each example for yourself. Whenever anew term is introduced, add it along with its definition to your glossary. Whenever a newtheorem is introduced, add it to your list of theorems.3.Try the problems for each section of unit 5. Try to avoid looking at the answers until you havemade every effort to solve the problem yourself.4.When you have completed the work for this unit, review the learning objectives for the unit andbe sure that you can do all of them. Review areas that you found to be difficult and try some ofthe problems again.
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