Problem 8 - Algebra and Trigonometry Real Mathematics, Real People

Question

What is the dimension of \(A B\) when \(A\) is a \(2 \times 3\) matrix and \(B\) is a \(3 \times 4\) matrix?

Step-by-Step Solution

Answer

The dimension of the matrix \(AB\) is \(2 \times 4\).

Step 1: Verify the multiplication condition

Verify that the number of columns in matrix \(A\) is equal to the number of rows in matrix \(B\). Since matrix \(A\) has 3 columns and matrix \(B\) has 3 rows, they can be multiplied.

Step 2: Determine the dimension of the product

The dimension of the product matrix \(AB\) is given by the number of rows of the first matrix \(A\) and the number of columns of the second matrix \(B\). Therefore, the dimension of \(AB\) is \(2 \times 4\).

Key Concepts

Dimensions of a Matrix Product

Understanding how to find the dimensions of a matrix product is crucial for performing matrix multiplication. The dimensions of the product result directly from the dimensions of the two matrices involved in the multiplication. Let's explore this concept using our example where matrix A is a 2 x 3 matrix and matrix B is a 3 x 4 matrix.

To determine the dimension of the product matrix, you simply take the number of rows of the first matrix, A, and the number of columns of the second matrix, B. So, if we follow this rule, the resulting product matrix, AB, will have dimensions of 2 x 4. This means the product matrix will have 2 rows and 4 columns, representing a grid of numbers with 2 horizontal rows and 4 vertical columns.

It's important to remember that only the 'outer' dimensions are considered when determining the size of the resulting matrix. The 'inner' dimensions (the columns of the first matrix and rows of the second) must match for multiplication to be possible but do not directly influence the size of the product.

Matrix Multiplication Conditions

Matrix multiplication isn't as straightforward as multiplying single numbers. Specific conditions must be met for two matrices to be multiplied together. The fundamental condition for matrix multiplication to be viable is that the number of columns in the first matrix must equal the number of rows in the second matrix.

In the provided exercise, matrix A has 3 columns, and matrix B has 3 rows. Since these numbers are equal, multiplication is possible—this is also known as the 'inner dimensions' matching. If they didn't match, say, if A had 3 columns and B had 5 rows, we would not be able to multiply them, and the concept of a resulting product matrix would be undefined. This rule ensures each element of the first matrix pairs with the corresponding element of the second matrix to produce the summed product entries in the resulting matrix.

Matrix Sizes

The size, or dimensions of a matrix, are described by the number of rows and columns it contains, typically denoted as m x n where m is the number of rows and n is the number of columns. The size of a matrix is fundamental in many matrix operations, including addition, subtraction, and especially multiplication.

For example, if we have a 2 x 3 matrix, this matrix consists of 2 rows and 3 columns. When performing operations that involve two or more matrices, it's crucial to be mindful of their sizes. Incompatible matrix sizes can lead to errors or undefined operations in the case of multiplication, as each element of a row from one matrix must have a corresponding element in a column of another matrix.

When discussing the dimensions of a matrix product, it's important to note that the result can have completely different dimensions compared to the original matrices involved. As seen in our exercise, a 2 x 3 matrix and a 3 x 4 matrix produce a 2 x 4 product, showcasing that the dimensions of matrix products can be quite diverse, depending on the sizes of the matrices being multiplied.