Question

To find the product $A B$ of two matrices $A$ and $B,$ which of the following must be true? (a) The number of columns in $A$ must equal the number of rows in $B$. (b) The number of rows in $A$ must equal the number of columns in $B$. (c) $A$ and $B$ must have the same number of rows and the same number of columns. (d) $A$ and $B$ must both be square matrices.

Name: Problem 7, Chapter 8: College Algebra
Uploaded: 2020-05-10T04:13:11-08:00
Duration: 1 min 35 s
Channel: Patrick Burns
Description: To find the product A B of two matrices A and B, which of the following must be true? (a) The number of columns in A must equal the number of rows in B. (b) The number of rows in A must equal the number of columns in B. (c) A and B must have the same number of rows and the same number of columns. (d) A and B must both be square matrices.

   To find the product $A B$ of two matrices $A$ and $B,$ which of the following must be true?
(a) The number of columns in $A$ must equal the number of rows in $B$.
(b) The number of rows in $A$ must equal the number of columns in $B$.
(c) $A$ and $B$ must have the same number of rows and the same number of columns.
(d) $A$ and $B$ must both be square matrices.

College Algebra

Michael Sullivan 10th Edition

Chapter 8, Problem 7

Instant Answer

Solved by Expert Patrick Burns

05/10/2020

Step 1

Let's say $A$ is of order $m \times n$ and $B$ is of order $p \times q$. This means $A$ has $m$ rows and $n$ columns, and $B$ has $p$ rows and $q$ columns. Show more…

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To find the product $A B$ of two matrices $A$ and $B,$ which of the following must be true? (a) The number of columns in $A$ must equal the number of rows in $B$. (b) The number of rows in $A$ must equal the number of columns in $B$. (c) $A$ and $B$ must have the same number of rows and the same number of columns. (d) $A$ and $B$ must both be square matrices.

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Key Concepts

Matrix Multiplication

Matrix multiplication is an operation that involves taking the rows of the first matrix and the columns of the second matrix and computing their dot products to form the entries of a new matrix. This process requires specific compatibility between the matrices involved, ensuring that the operation is defined and meaningful.

Dimensional Compatibility for Multiplication

For the multiplication of two matrices A and B to be defined, the number of columns in matrix A must equal the number of rows in matrix B. This is the fundamental condition that allows each element of the product matrix to be computed as the sum of products of corresponding elements from the rows of A and the columns of B.

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