Determinants and Cramer's Rule

This lecture covers the calculation of determinants by expanding along a row or column and performing elementary row operations, illustrated with examples. It then introduces Cramer's Rule for solving systems of linear equations using determinants, along with the formula for the matrix inverse derived from the rule. The importance and application of Cramer's Rule are discussed, emphasizing its utility in studying solution dependency. The lecture concludes with the concept of vector spaces, defining vector addition and scalar multiplication operations, and presenting the ten axioms that characterize a vector space.