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Chapter 16: Problem 13

What do you need to know about two matrices to know if their product exists?

Short Answer

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Question: Explain the conditions for matrix multiplication to exist and provide an example to illustrate the concept. Answer: For the product of two matrices A and B to exist, the number of columns in Matrix A must be equal to the number of rows in Matrix B. In other words, if Matrix A has dimensions m x n (m rows and n columns), and Matrix B has dimensions p x q (p rows and q columns), then matrix multiplication can be performed if n = p. The resulting matrix C will have dimensions m x q (m rows and q columns). For instance, consider Matrix A has dimensions 2x3 and Matrix B has dimensions 3x4. Since the number of columns in Matrix A (3) is equal to the number of rows in Matrix B (3), their product exists, and the resulting matrix C will have dimensions 2x4.

Step by step solution

01

Understanding Dimensions of Matrices

First, remember the dimensions of a matrix are represented in the form of rows x columns. For example, a matrix with dimensions 3x2 has 3 rows and 2 columns.
02

Determine the Dimensions of the Matrices

Let's denote the dimensions of the two matrices as A and B: - Matrix A has dimensions m x n (m rows and n columns) - Matrix B has dimensions p x q (p rows and q columns)
03

Check the Condition for Matrix Multiplication

To know if the product of two matrices A and B exists, the number of columns in Matrix A (n) should be equal to the number of rows in Matrix B (p). In this case, n = p.
04

Find the Dimension of the Resultant Matrix

If the product of the two matrices exists, the resultant matrix C will have dimensions m x q (m rows and q columns). This is because the number of rows comes from matrix A and the number of columns comes from matrix B. For example: Matrix A has dimensions 2x3 (2 rows and 3 columns) Matrix B has dimensions 3x4 (3 rows and 4 columns) Since the number of columns in Matrix A (3) is equal to the number of rows in Matrix B (3), the product exists. The dimension of the resultant matrix C is 2x4 (2 rows and 4 columns).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Matrix Dimensions
When dealing with matrices, one of the first things to understand is the concept of dimensions. The dimensions of a matrix are expressed as the number of rows followed by the number of columns, and it's usually written in the form \( m \times n \). For example, a matrix with 3 rows and 2 columns has the dimensions 3x2. This information is crucial because it defines the matrix's structure.
  • The first value in the dimensions (\( m \)) represents the number of horizontal lines, or rows.
  • The second value (\( n \)) represents the vertical lines, or columns.
Understanding matrix dimensions is the foundation for learning about more advanced operations, such as matrix multiplication. By knowing the dimensions, you can determine how various matrices can interact with each other.
Matrix Product Existence
The existence of a matrix product, when multiplying two matrices, depends on a specific dimension condition being met. This condition revolves around the relationship between rows and columns of the two matrices involved in the multiplication. To multiply two matrices, the following must be true:
  • The number of columns in the first matrix \( (n) \) must equal the number of rows in the second matrix \( (p) \).
If this condition is satisfied, then the matrices can be multiplied. If not, the multiplication simply cannot occur. This "inner dimension" must match because it ensures that the elements of the matrices line up properly for multiplication. For example, if matrix A is \( 2x3 \) and matrix B is \( 3x4 \), the inner dimensions (3 in A and 3 in B) match, so the product can be computed.
Resultant Matrix Dimensions
Once you have determined that the product of two matrices can exist by checking the dimensions, the next step is to find out the dimensions of the resultant matrix. The dimensions of the resulting matrix from a matrix multiplication are determined by the "outer dimensions" of the original matrices:
  • The number of rows in the resultant matrix comes from the first matrix \( (m) \).
  • The number of columns in the resultant matrix comes from the second matrix \( (q) \).
For instance, if matrix A has dimensions \( 2x3 \) and matrix B has dimensions \( 3x4 \), the resulting matrix, after multiplication, will have dimensions \( 2x4 \). This is because the outer dimensions (2 from A and 4 from B) direct the size of the resulting product matrix. Understanding how to predict the size of the resultant matrix can significantly simplify matrix operations and is a core concept in linear algebra.

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Most popular questions from this chapter

Find a single vector resulting from the operations. \(\frac{1}{2}\left(\left(\frac{1}{7}, 3,4\right)+\left(3, \frac{1}{2}, 9\right)\right)\)

Check the statements in Exercises \(19-23\) using the matrices \(\mathbf{U}=\left(\begin{array}{ll}2 & 3 \\ 1 & 2\end{array}\right), \mathbf{V}=\left(\begin{array}{cc}-1 & 4 \\ 0 & 2\end{array}\right), \mathbf{W}=\left(\begin{array}{cc}5 & -5 \\ 4 & 7\end{array}\right).\) $$ (\mathbf{W}+\mathbf{U})+\mathbf{V}=\mathbf{W}+(\mathbf{U}+\mathbf{V}) $$

If possible, use row operations to solve the systems. $$ \left\\{\begin{array}{l} -13 x+26 y=-13 \\ 3 x-8 y=-1 \end{array}\right. $$

Write the system of equations described by the augmented matrices. $$ \left(\begin{array}{ccc|c} 1 & 2 & 3 & 4 \\ 5 & 6 & 7 & 8 \\ 9 & 10 & 11 & 12 \end{array}\right) $$

Find a single vector resulting from the operations. \(\left(\begin{array}{cc}5 & 10 \\ 20 & 40\end{array}\right)\left(\begin{array}{l}3 \\ 7\end{array}\right)\)
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