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Chapter 3: Problem 77

Give an example of two matrices \(A\) and \(B\) such that \(A B\) is defined but \(B A\) is not defined.

Short Answer

Expert verified
An example of two matrices $A$ and $B$ such that $AB$ is defined but $BA$ is not defined are: Matrix A (2 x 3): \[\begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix}\] Matrix B (3 x 1): \[\begin{pmatrix} 7 \\ 8 \\ 9 \end{pmatrix}\] The product $AB$ is defined as \[\begin{pmatrix} 50 \\ 122 \end{pmatrix}\], but the product $BA$ is undefined because the number of columns of B (1) is not equal to the number of rows of A (2).

Step by step solution

01

Choose the dimensions for matrices A and B

Let's choose matrix A as an m x n matrix, and matrix B as an n x p matrix. For our purposes, we can choose the dimensions so that the product AB is defined but BA is not. We can pick m = 2, n = 3, and p = 1.
02

Construct the matrices

With the chosen dimensions (m = 2, n = 3, p = 1), let's create the matrices A and B: Matrix A (2 x 3): \[\begin{pmatrix} a & b & c \\ d & e & f \end{pmatrix}\] Matrix B (3 x 1): \[\begin{pmatrix} g \\ h \\ i \end{pmatrix}\]
03

Choose specific values for the elements

Now let's choose some specific values for the elements of matrices A and B. To make it simple, let's just use integers in the range from 1 to 9: Matrix A (2 x 3): \[\begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix}\] Matrix B (3 x 1): \[\begin{pmatrix} 7 \\ 8 \\ 9 \end{pmatrix}\]
04

Show that the product AB is defined

The product of matrices A and B is defined because the number of columns of A (3) is equal to the number of rows of B (3). The product AB will be a 2x1 matrix, which it indeed is: \[AB = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix}\begin{pmatrix} 7 \\ 8 \\ 9 \end{pmatrix} = \begin{pmatrix} 1\cdot7 + 2\cdot8 + 3\cdot9 \\ 4\cdot7 + 5\cdot8 + 6\cdot9 \end{pmatrix} = \begin{pmatrix} 50 \\ 122 \end{pmatrix}\]
05

Show that the product BA is undefined

The product B multiplied by A is undefined because the number of columns of B (1) is not equal to the number of rows of A (2), as we can observe from the dimensions of the matrices: Matrix B (3 x 1): \[\begin{pmatrix} 7 \\ 8 \\ 9 \end{pmatrix}\] Matrix A (2 x 3): \[\begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix}\] The product BA is undefined because the dimensions do not agree, meaning that the matrix product operation cannot be performed.

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