Problem 27 - Calculus for Biology and Medicine

Question

Suppose that \(A\) is a \(3 \times 4\) matrix and \(B\) is a \(4 \times 2\) matrix. What is the size of the product \(A B ?\)

Step-by-Step Solution

Answer

The size of the product matrix \(AB\) is \(3 \times 2\).

Step 1: Understanding Matrix Multiplication

When multiplying two matrices, the number of columns in the first matrix must be equal to the number of rows in the second matrix. In this case, matrix \(A\) has 3 rows and 4 columns, while matrix \(B\) has 4 rows and 2 columns. Since the number of columns in \(A\) is equal to the number of rows in \(B\), matrix multiplication is possible.

Step 2: Determine the Size of the Product Matrix

The size of the resulting matrix from the product of two matrices is determined by the number of rows of the first matrix and the number of columns of the second matrix. Therefore, the matrix \(AB\) will have the same number of rows as matrix \(A\) (which is 3) and the same number of columns as matrix \(B\) (which is 2).

Key Concepts

Matrix Dimensions

Matrix dimensions are a fundamental aspect of matrices, representing the size of a matrix in terms of its rows and columns. Each matrix is described in terms of "rows × columns." For example, if you have a matrix labeled as a "3 × 4" matrix, this means it has:

• 3 rows, which are horizontal line-ups of numbers.
• 4 columns, which are vertical stacks of numbers.

Understanding these dimensions is crucial when dealing with matrices, especially when considering operations like addition and multiplication. The order of rows and columns, in this balanced structure, affects how matrices interact with each other. So, always pay close attention to the number of rows and columns indicated in a matrix's description.
By knowing the dimensions, you can quickly assess what operations can be performed on or between matrices.

Compatible Matrices

When it comes to multiplying matrices, not all matrix pairings are compatible. For two matrices to be multiplied, the number of columns in the first matrix must match the number of rows in the second matrix.
For instance, in the exercise you are considering, matrix \(A\) is 3 × 4 and matrix \(B\) is 4 × 2. The 4 columns in matrix \(A\) align perfectly with the 4 rows in matrix \(B\), making them compatible for multiplication.

• This matching condition ensures a proper "dot product" computation between matrices.
• If the inner dimensions (4 in our example) do not match, multiplication is not possible.

Thus, understanding matrix compatibility helps avoid mistakes when planning or executing matrix operations. Compatibility ensures that the matrix multiplication process is mathematically sound and results in a meaningful product.

Product Matrix Size

Once we've determined that two matrices can indeed be multiplied, the next important step is to find out the size of the result, known as the product matrix. The dimensions of this product matrix are determined by the outer dimensions—the number of rows of the first matrix and the number of columns of the second matrix.
If matrix \(A\) has dimensions \(3 \times 4\) and matrix \(B\) has dimensions \(4 \times 2\), then the product matrix \(AB\) will have:

• The same number of rows as \(A\), which is 3.
• The same number of columns as \(B\), which is 2.

This results in the product matrix \(AB\) being a \(3 \times 2\) matrix.
Knowing how to determine the product matrix size is essential for organizing your matrix work and predicting the layout of results. This understanding equips you to handle more complex matrix operations with ease and accuracy.