Problem 23 - Algebra 2

Question

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

Step-by-Step Solution

Answer

The product HF is undefined, as the number of columns in H (1) does not match the number of rows in F (2).

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. For matrix multiplication to be defined for matrices A and B (where A is an m x n matrix and B is a p x q matrix), it must be the case that n = p.

Step 2: Determining the size of matrices H and F

First, determine the size of matrix H, which is a 2 x 1 matrix since it has 2 rows and 1 column. Matrix F is a 2 x 2 matrix with 2 rows and 2 columns.

Step 3: Checking the compatibility of H and F for multiplication

Since matrix H has one column and matrix F has two rows, the number of columns in H (1) matches the number of rows in F (2), which makes the product HF undefined because the number of rows in matrix H must match the number of columns in matrix F.

Key Concepts

Undefined Matrix Product

When learning about matrices, an important concept to understand is when a matrix product is undefined. In the context of the given exercise, we consider the product of matrices H and F. You’ll remember that the matrix product is defined only when the matrices involved can be comfortably multiplied according to specific rules.

Let's look at the size of matrices H and F. Matrix H is a 2 x 1 matrix and matrix F is a 2 x 2 matrix. To determine whether multiplication is possible, we follow the rule that says the number of columns in the first matrix (Matrix H, which has 1 column) must equal the number of rows in the second matrix (Matrix F, which has 2 rows). Since this condition isn't met in our case, we say that the product HF is undefined.

Why does this matter? Understanding when a matrix product is undefined helps to avoid errors in calculations and ensures you're working with compatible matrices. It's akin to ensuring puzzle pieces fit together; if they don't, you cannot successfully complete the puzzle. Similarly, if the matrices are incompatible, the multiplication cannot be performed.

Matrix Dimensions

In order to fully grasp the concept of matrix multiplication, one must be familiar with matrix dimensions. The dimensions of a matrix are expressed as 'rows x columns'. For example, consider the given matrix H which is expressed as a 2 x 1 matrix because it has 2 rows and 1 column. Similarly, matrix F is a 2 x 2 matrix with its 2 rows and 2 columns.

Knowing the dimensions of a matrix is not merely an exercise in counting; it is crucial for various matrix operations, including addition, subtraction, and multiplication. In multiplication, the dimensions directly determine whether two matrices can be multiplied, as they must conform to the rule of matrix compatibility. Matrix dimensions serve as a guideline and provide the structure necessary for matrix operations to function logically. Remember, just as you need to know the measurements to cut pieces of wood to build a chair, you need to know the dimensions of your matrices to combine them in meaningful ways.

Matrix Compatibility

Matrix compatibility is a key concept to understand when dealing with matrix multiplication. Two matrices are considered compatible for multiplication if, and only if, the number of columns in the first matrix is equal to the number of rows in the second matrix.

Real-World Analogy:

Think of it as a lock and key system. Just as the right key has to have the correct pattern to fit into a lock, the matrix on the left must have the same number of columns as the number of rows in the matrix on the right for them to multiply.

In our exercise, matrix H cannot be multiplied by matrix F because their dimensions are not compatible: H has 1 column and F has 2 rows. Thus, the lock doesn't match the key. It’s important to check for matrix compatibility before starting any calculation to save time and effort on operations that cannot be performed. Always remember to compare the right dimension of the first matrix to the left dimension of the second one to ensure compatibility.