Guided Proof Prove Theorem 3.9: If is a square matrix, then det Getting Started: To

Chapter 3, Problem 56

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Guided Proof Prove Theorem 3.9: If is a square matrix, then det Getting Started: To prove that the determinants of and are equal, you need to show that their cofactor expansions are equal. Because the cofactors are determinants of smaller matrices, you need to use mathematical induction. (i) Initial step for induction: If is of order 1, then so (ii) Assume the inductive hypothesis holds for all matrices of order Let be a square matrix of order Write an expression for the determinant of by expanding by the first row. (iii) Write an expression for the determinant of by expanding by the first column. (iv) Compare the expansions in (i) and (ii). The entries of the first row of are the same as the entries of the first column of Compare cofactors (these are the determinants of smaller matrices that are transposes of one another) and use the inductive hypothesis to conclude that they are equal as well.