Problem 27 - Algebra 2

Question

Determine whether each product is defined or undefined. $$F=\left[\begin{array}{ll}{2} & {3} \\ {6} & {9}\end{array}\right] \quad G=\left[\begin{array}{rr}{-3} & {6} \\ {2} & {-4}\end{array}\right] \quad H=\left[\begin{array}{r}{-5} \\ {6}\end{array}\right] \quad J=\left[\begin{array}{ll}{0} & {7}\end{array}\right]$$ $$ J F $$

Step-by-Step Solution

Answer

The product JF is defined and will result in a 1x2 matrix.

Step 1: Understanding Matrix Multiplication

Matrix multiplication is defined when the number of columns in the first matrix is equal to the number of rows in the second matrix. In this case, to determine if the product of matrices J and F is defined, we need to check if the number of columns in J equals the number of rows in F.

Step 2: Checking the Dimensions of Matrices J and F

Matrix J has dimensions 1x2 (1 row and 2 columns) and matrix F has dimensions 2x2 (2 rows and 2 columns). Since the number of columns in J (which is 2) is equal to the number of rows in F (which is also 2), the product JF is defined.

Step 3: Describing the Product

The product of matrices J and F (JF) will result in a new matrix with the dimensions of J's rows by F's columns, which will be a 1x2 matrix.

Key Concepts

Defined Matrix Product

Understanding the rules for matrix multiplication is crucial for anyone delving into linear algebra or related fields. A matrix product is defined only when specific conditions about the matrix dimensions are met. Specifically, the number of columns in the first matrix must be equal to the number of rows in the second matrix.

For example, consider matrices J and F from the exercise. The matrix J is a 1x2 matrix, and F is a 2x2 matrix. The compatibility rule allows us to multiply J and F because J’s 2 columns match F’s 2 rows. Hence, the matrix product JF is defined. The outcome of this multiplication will be a matrix whose dimensions are determined by the outer dimensions of the matrices involved. Since J has 1 row and F has 2 columns, the resulting matrix from JF will be 1x2.

Matrix Dimensions

The concept of matrix dimensions, expressed as 'm x n', where 'm' indicates the number of rows and 'n' signifies the number of columns, is fundamental when dealing with matrix operations, including multiplication.

Using the exercise as context, it’s essential to examine the dimensions of the matrices before proceeding with any operations. The dimensions not only guide us in understanding whether a matrix product is defined but also predict the size of the resulting matrix after multiplication. The product's dimensions will be the number of rows of the first matrix by the number of columns of the second matrix. Understanding this helps avoid common errors when performing matrix multiplication.

Additionally, this knowledge assists in grasping more complex applications of matrices, such as transformations in graphics or solving systems of linear equations.

Algebraic Structures

Matrices are integral components of algebraic structures in mathematics, particularly in linear algebra. An algebraic structure consists of a set equipped with one or more operations that combine its members according to certain rules. With matrices, these operations include addition, subtraction, multiplication, and scalar multiplication.

Understanding matrix multiplication within the context of algebraic structures enhances comprehension of how matrices can be used to represent and solve numerous mathematical problems. It provides a structured approach to examining mathematical entities and their interactions. For example, in systems of linear equations, matrices represent the coefficients and constants that can be manipulated to find solutions, demonstrating the practical application of these algebraic structures in both theoretical and real-world situations.