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In Exercises 31–36, mention an appropriate theorem in your explanation.Let $A$ and $B$ be square matrices. Show that even though $A B$ and $B A$ may not be equal, it is always true that det $A B=\operatorname{det} B A$

see the proof

Algebra

Chapter 3

Determinants

Section 2

Properties of Determinants

Introduction to Matrices

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were given a statement and were asked to prove this statement using the theorems from the section. So let A and B B square matrices and were asked to show that even though multiplication is not necessarily communicative so that a B is not necessarily equal to be a it is always true that the determinant of a B is equal to the determinant of the matrix B A no, we have since both and beer square matrices It follows that the major sees a bee and be a are also square matrices. And so the determinant is defined and by the're, um six we have of the determinant of a B is equal to the determinant of a times the determinant of B which is equal to since scaler zehr communicative determinant of b times the determinant of a and then again by the're, um six. This is the determinant of B A. So we see that these two determinants are

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