Problem 43 - A Graphical Approach to Precalculus with Limits

Question

The dimensions of matrices \(A\) and \(B\) are given. Find the dimensions of the product \(A B\) and of the product BA if the products are defined. If they are not defined, say so. $$A \text { is } 3 \times 5 ; B \text { is } 5 \times 2$$

Step-by-Step Solution

Answer

AB is defined with dimensions 3x2; BA is not defined.

Step 1: Verify Matrix Dimensions for AB

For the matrix product \(AB\) to be defined, the number of columns in matrix \(A\) must be equal to the number of rows in matrix \(B\). Here, \(A\) is \(3 \times 5\) and \(B\) is \(5 \times 2\). Since both have matching numbers of columns and rows (both 5), the product \(AB\) is defined.

Step 2: Determine Dimensions of AB

The dimensions of the resulting matrix from the product \(AB\) will be given by the number of rows in \(A\) and the number of columns in \(B\). Therefore, the dimensions of \(AB\) are \(3 \times 2\).

Step 3: Verify Matrix Dimensions for BA

For the matrix product \(BA\) to be defined, the number of columns in matrix \(B\) must be equal to the number of rows in matrix \(A\). Here, \(B\) is \(5 \times 2\) and \(A\) is \(3 \times 5\). Since the number of columns in \(B\) (2) does not match the number of rows in \(A\) (3), the product \(BA\) is not defined.

Key Concepts

Matrix Dimensions

Understanding the dimensions of a matrix is crucial when performing matrix operations, especially multiplication. A matrix is simply an arrangement of numbers into rows and columns. The size or dimension of a matrix is given in terms of rows and columns. For example, a matrix with 3 rows and 5 columns is called a "3 by 5" matrix, noted as \(3 \times 5\).

When multiplying two matrices, the key rule to remember is that the number of columns in the first matrix must match the number of rows in the second matrix. This compatibility rule ensures that each row from the first matrix has a corresponding column in the second matrix to compute the product.

• For a matrix \(A\) with dimensions \(3 \times 5\) and matrix \(B\) with dimensions \(5 \times 2\), the product \(AB\) is possible because the columns of \(A\) match the rows of \(B\).
• Conversely, if \(B\) has 2 columns and you tried to multiply it with \(A\) having 3 rows \((BA)\), the operation is not defined due to a mismatch.

Matrix Product

The result of matrix multiplication, known as the matrix product, follows specific rules. When \(AB\) is formed, each element in the resulting matrix is calculated by taking a row from \(A\) and a column from \(B\), then computing the sum of products of their corresponding elements. This method is known as the dot product.

The dimensions of the resulting product matrix depend on the outer dimensions of the original matrices. For matrices \(A\) with dimensions \(3 \times 5\) and \(B\) with dimensions \(5 \times 2\), the product \(AB\) will be a matrix with dimensions \(3 \times 2\). It will have:

• 3 rows corresponding to the rows of \(A\).
• 2 columns corresponding to the columns of \(B\).

This size rule helps determine the dimensions of any resultant matrix without performing actual multiplication.

Linear Algebra

Matrix multiplication is a fundamental concept in linear algebra, a branch of mathematics dealing with vectors, matrices, and linear transformations. In this domain, matrices often represent systems of equations or transformations of space.

Understanding matrix dimensions and products is critically important in applications like computer graphics, where matrices are used to change the perspective and rotation of images or objects through transformations. It's also crucial in solving systems of linear equations, where matrices streamline processes that would be cumbersome with basic algebra.

• In linear transformations, as you multiply matrices, you often alter the direction or shape of data in a coordinate plane or space.
• In solving linear systems, multiplying matrices aligns with obtaining solutions through methods like Gaussian elimination or using inverses.

Thus, mastering matrix multiplication unlocks vast applications in fields that rely significantly on mathematical modeling and computational simulations.