Problem 42 - A Graphical Approach to College Algebra

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. $$A \text { is } 3 \times 1 ; B \text { is } 1 \times 3.$$

Step-by-Step Solution

Answer

AB is a 3x3 matrix; BA is a 1x1 matrix.

Step 1: Understand Matrix Dimensions

Matrix \(A\) is given as \(3 \times 1\), which means it has 3 rows and 1 column. Matrix \(B\) is \(1 \times 3\), meaning it has 1 row and 3 columns.

Step 2: Check Conditions for Matrix Multiplication

For the matrix product \(AB\) to be defined, the number of columns in \(A\) (which is 1) must equal the number of rows in \(B\) (which is 1). Since these dimensions match, \(AB\) is defined.

Step 3: Determine Dimensions of Product AB

The resulting matrix from the multiplication \(AB\) will have dimensions of the number of rows of \(A\) by the number of columns of \(B\). Therefore, \(AB\) is a \(3 \times 3\) matrix.

Step 4: Check Conditions for Matrix Multiplication BA

For the matrix product \(BA\) to be defined, the number of columns in \(B\) (which is 3) must match the number of rows in \(A\) (which is 3). These dimensions match, so \(BA\) is defined.

Step 5: Determine Dimensions of Product BA

The resulting matrix from the multiplication \(BA\) will have dimensions of the number of rows of \(B\) by the number of columns of \(A\). Therefore, \(BA\) is a \(1 \times 1\) matrix.

Key Concepts

Matrix Dimensions

Understanding matrix dimensions is crucial when working with matrices. A matrix is essentially an array of numbers organized in rows and columns.
Each matrix has dimensions represented by the number of rows and columns it contains, often written as "rows × columns".

• If a matrix has 3 rows and 1 column, like matrix \(A\) in our example, it's a \(3 \times 1\) matrix.
• Similarly, if a matrix has 1 row and 3 columns, like matrix \(B\), it is a \(1 \times 3\) matrix.

Understanding the size and shape of matrices allows you to determine how operations, like multiplication, can be performed between them. Each dimension gives you a visual layout of your matrix: columns by rows or vice versa. Knowing this is the first step in performing operations such as addition, subtraction, and particularly multiplication.

Matrix Products

The product of two matrices, known as matrix multiplication, results in a new matrix. But not all matrices can be multiplied with each other.
To perform matrix multiplication, certain conditions must be met (which we'll discuss in detail in the next section). Matrix multiplication isn't as straightforward as multiplying numbers.
When you multiply matrices, you essentially take the dot product of rows and columns.

• For the product \(AB\), you'd align the rows of matrix \(A\) with the columns of matrix \(B\), ensuring compatibility in dimensions.
• The resulting product \(AB\) has dimensions based on the rows of "A" and the columns of "B". In our example, that's a \(3 \times 3\) matrix.
• Similarly, the product \(BA\) would have dimensions based on the rows of "B" and the columns of "A", resulting in a \(1 \times 1\) matrix in our case.

Matrix products create new matrices, and the resulting sizes depend primarily on the initial matrices' dimensions.

Matrix Multiplication Conditions

Before attempting to multiply two matrices, you must verify that multiplication is possible. This is determined particularly by their inner dimensions.
When we say inner dimensions, we mean the number of columns of the first matrix and the number of rows of the second matrix. These must be equal for the multiplication to be valid.

• In the example provided, matrix \(A\) \((3 \times 1)\) has 1 column, which matches the 1 row in matrix \(B\) \((1 \times 3)\), so the product \(AB\) is defined.
• Conversely, for the product \(BA\) \((1 \times 3)\) \(\times\) \((3 \times 1)\), the 3 columns in \(B\) match the 3 rows in \(A\), therefore \(BA\) is also defined.

These conditions are essential for identifying whether a product such as \(AB\) or \(BA\) is possible. If the inside dimensions don't match, it’s like trying to fit a square peg into a round hole; the multiplication simply can't happen.