In this exercise, we prove the determinantal product formula...

In this exercise, we prove the determinantal product formula (1.85). (a) Prove that if $E$ is any elementary matrix (of the appropriate size), then $\operatorname{det}(E B)=\operatorname{det} E \operatorname{det} B$. (b) Use induction to prove that if $A=E_1 E_2 \cdots E_N$ is a product of elementary matrices, then $\operatorname{det}(A B)=\operatorname{det} A \operatorname{det} B$. Explain why this proves the product formula whenever $A$ is a nonsingular matrix. (c) Prove that if $Z$ is a matrix with a zero row, then $Z B$ also has a zero row, and so $\operatorname{det}(Z B)=0=\operatorname{det} Z \operatorname{det} B$. (d) Use Exercise 1.8.21 to complete the proof of the product formula.

Key Concepts

Proof by Induction

Proof by induction is a fundamental mathematical technique for establishing the validity of a statement for all natural numbers. In the context of this exercise, induction is used to prove that the product formula for determinants holds for any product of a finite number of elementary matrices. This method helps extend results from a base case to more general cases in a logical and structured manner.

Zero Row Property

The zero row property in determinants states that if a matrix has a row (or column) that is entirely zero, then its determinant is zero. This property is essential in proving the determinant product formula in scenarios where one of the matrices involved, or their product, may contain a zero row. Recognizing the persistence of a zero row in the matrix product reinforces the multiplicative property of the determinant.

Nonsingular Matrices

A nonsingular matrix (or invertible matrix) is one that has a nonzero determinant and therefore an inverse. The importance of nonsingular matrices in this context comes from the fact that any such matrix can be decomposed into a product of elementary matrices. This factorization is leveraged in proofs and other arguments in linear algebra, as it allows properties of determinants to be transferred from simple cases (elementary matrices) to more complex matrices.

Determinantal Product Formula

This concept refers to the fundamental property in linear algebra that the determinant of the product of two square matrices is equal to the product of their determinants. It is a key result that connects the algebraic properties of matrices with the geometric interpretation of volume scaling, and it underpins many other results and methods in matrix theory.

Elementary Matrices

Elementary matrices are matrices obtained by performing a single elementary row (or column) operation on an identity matrix. They are crucial because they represent the basic transformations possible on matrices, and any invertible (nonsingular) matrix can be expressed as a product of such matrices. Their determinant properties, particularly that the determinant of an elementary matrix is easy to compute, are essential in proving more complex results like the determinantal product formula.

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