Problem 29 - College Algebra

Question

In Exercises \(27-36,\) find (if possible): \(\begin{array}{llll}\text { a. } A B & \text { and } & \text { b. } B A\end{array}\) $$ A=\left[\begin{array}{llll} 1 & 2 & 3 & 4 \end{array}\right], \quad B=\left[\begin{array}{l} 1 \\ 2 \\ 3 \\ 4 \end{array}\right] $$

Step-by-Step Solution

Answer

The product of the matrices \(AB = 30\), a \(1x1\) matrix. The product \(BA\) is not possible to calculate due to incompatible matrix dimensions.

Step 1: Calculate \(AB\)

To find the product \(AB\), multiply each element in the row of matrix \(A[1, 2, 3, 4]\) with the corresponding element in the column of matrix \(B[1, 2, 3, 4]^T\) and add them up. This gives us \(AB = 1*1 + 2*2 + 3*3 + 4*4 = 30\). The resulting matrix \(AB\) is a \(1x1\) matrix with the only element \(30\).

Step 2: Determine if \(BA\) is possible

The product \(BA\) would result in a matrix where each element represents the sum of the product of elements in the row of the first matrix \(B[1, 2, 3, 4]^T\) and the column of the second matrix \(A[1, 2, 3, 4]\). However, \(B\) only has one column and \(A\) only has one row, making \(BA\) impossible to calculate.

Step 3: Conclusion

Therefore, the product of matrices \(AB\) is possible and equals a \(1x1\) matrix with element \(30\), while the product \(BA\) is not possible due to incompatible matrix dimensions.

Key Concepts

Matrices

Matrices are a fundamental concept in mathematics, specifically in the field of linear algebra. They are rectangular arrays of numbers, symbols, or expressions arranged in rows and columns. Each individual entry in a matrix is known as an element. Matrices are not just a mathematical construct; they have practical applications in different areas such as physics, engineering, computer science, and economics.

For example, in the context of the given exercise, matrix A is a representation of a single row with four elements and matrix B is a single column with four elements. These matrices can contain any real numbers or variables and can be manipulated according to the rules of matrix arithmetic.

One crucial operation involving matrices is multiplication, where elements of the rows and columns are combined in a specific manner to produce a new matrix. Understanding how to correctly multiply matrices allows one to solve various types of problems in science and engineering that relate to linear transformations and systems of equations.

Linear Algebra

Linear algebra is the branch of mathematics concerning vector spaces and linear mappings between such spaces. It includes the study of lines, planes, and subspaces but is also concerned with properties common to all vector spaces. Matrices are one of the key tools used in linear algebra because they can represent linear transformations and systems of linear equations.

In the context of our example, when we compute matrix AB, we are, in essence, applying a linear transformation to a vector represented by matrix B using the transformation matrix A. This process is fundamental to understanding and solving linear equations, transforming geometric objects, and handling data in computer algorithms.
If we were to consider additional problems, linear algebra would assist in determining whether given vectors are linearly independent, in finding eigenvalues and eigenvectors, and in many other sophisticated applications. It's a powerful framework with which one can approach and solve a wide array of mathematical problems.

Matrix Dimensions

Matrix dimensions play a crucial role in determining the operations that can be performed with matrices, such as addition, subtraction, and particularly, multiplication. The dimensions of a matrix are represented by the number of rows followed by the number of columns, often notated as 'm x n'.

For matrices to be multiplied, the number of columns in the first matrix must match the number of rows in the second matrix. In the example given, matrix A is a 1 x 4 matrix and matrix B is a 4 x 1 matrix. Therefore, the multiplication AB is possible since the number of columns of A (which is 4) matches the number of rows in B (which is 4). However, the attempted multiplication BA fails because the number of columns in B (1) does not match the number of rows in A (1).

In essence, matrix dimensions must align appropriately for multiplication; if they don't, this operation is undefined. Understanding this aspect of matrix dimensions is critical not just for performing calculations but also for conceptualizing the relationships between different vector spaces.