In Exercises 9 and 10, mark each statement True or False....

In Exercises 9 and $10,$ mark each statement True or False. Justify each answer. a. In order for a matrix $B$ to be the inverse of $A,$ both equations $A B=I$ and $B A=I$ must be true. b. If $A$ and $B$ are $n \times n$ and invertible, then $A^{-1} B^{-1}$ is the inverse of $A B .$ c. If $A=\left[\begin{array}{ll}{a} & {b} \\ {c} & {d}\end{array}\right]$ and $a b-c d \neq 0,$ then $A$ is invertible. d. If $A$ is an invertible $n \times n$ matrix, then the equation $A \mathbf{x}=\mathbf{b}$ is consistent for $\operatorname{each} \mathbf{b}$ in $\mathbb{R}^{n}$ . e. Each elementary matrix is invertible.

In Exercises 9 and $10,$ mark each statement True or False. Justify each answer.
a. In order for a matrix $B$ to be the inverse of $A,$ both equations $A B=I$ and $B A=I$ must be true.
b. If $A$ and $B$ are $n \times n$ and invertible, then $A^{-1} B^{-1}$ is the inverse of $A B .$
c. If $A=\left[\begin{array}{ll}{a} & {b} \\ {c} & {d}\end{array}\right]$ and $a b-c d \neq 0,$ then $A$ is invertible.
d. If $A$ is an invertible $n \times n$ matrix, then the equation $A \mathbf{x}=\mathbf{b}$ is consistent for $\operatorname{each} \mathbf{b}$ in $\mathbb{R}^{n}$ .
e. Each elementary matrix is invertible.

Key Concepts

Elementary Matrices

Elementary matrices are obtained by performing a single elementary row operation on the identity matrix. Each elementary matrix is invertible because these row operations are reversible; the inverse of an elementary matrix corresponds to the elementary row operation that reverses the effect of the original operation. These matrices are fundamental tools in understanding matrix manipulations and in the process of finding inverses.

Consistency of Linear Systems with Invertible Matrices

If A is an invertible n×n matrix, then the linear system A x = b has a unique solution for every vector b in R?. This property is a direct consequence of the invertibility of A, which guarantees that the mapping defined by A is both one-to-one and onto, ensuring the existence and uniqueness of solutions in linear systems.

Inverse of a Product

For any two invertible matrices A and B, the inverse of their product is given by (AB)?¹ = B?¹A?¹. This property highlights the importance of the order of multiplication in matrix products because, in general, matrices do not commute. Understanding the correct order is essential when solving problems involving inverses of products.

Determinant and Invertibility

For a square matrix, invertibility is equivalent to having a non-zero determinant. In the case of a 2×2 matrix, the standard determinant is computed as ad - bc. This criterion ensures that the matrix has full rank and that a unique inverse exists. Misstatements or errors in computing the determinant can lead to incorrect conclusions about a matrix’s invertibility.

Matrix Inverse Definition

A matrix B is said to be the inverse of a square matrix A if and only if both the product AB and BA yield the identity matrix. This means that the inverse operation ‘undoes’ the effect of the matrix multiplication by A and is unique when it exists. Both equations AB = I and BA = I are necessary conditions to establish that B is the true inverse of A.

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