Problem 11 - Contemporary Precalculus

Question

Evaluate the expression. $$\left(\begin{array}{l}4 \\\1\end{array}\right)\left(\begin{array}{l}5 \\\3\end{array}\right)$$

Step-by-Step Solution

Answer

Question: Evaluate the expression for the given matrices: $$\left(\begin{array}{l}4 \\\1\end{array}\right)\left(\begin{array}{l}5 \\\3\end{array}\right)$$ Answer: The result of the matrix multiplication is 23.

Step 1: Understand the Matrix multiplication

In order to multiply two matrices, the number of columns in the first matrix must match the number of rows in the second matrix. In this case, both matrices are 2x1, meaning they have 2 rows and 1 column. Therefore, we can proceed with the matrix multiplication.

Step 2: Multiply the corresponding entries

Multiply each entry of the first matrix with its corresponding entry in the second matrix and sum the results.

Step 3: Evaluate the expression

Evaluate the expression using matrix multiplication: $$\left(\begin{array}{l}4 \\\1\end{array}\right)\left(\begin{array}{l}5 \\\3\end{array}\right) = (4\cdot5) + (1\cdot3) = 20 + 3 = 23$$ We now have the result of the expression: $$\boxed{23}$$

Key Concepts

Matrix Dimensions

Before diving into matrix multiplication, it's essential to understand the dimensions of the matrices involved. The dimensions of a matrix are given in terms of rows and columns. For example, a matrix with 2 rows and 1 column is described as a 2x1 matrix.
Understanding dimensions is crucial because for two matrices to be multiplied, the number of columns in the first matrix must match the number of rows in the second matrix.
This rule ensures that each element in a row of the first matrix has a corresponding element in a column of the second matrix to multiply with. In our exercise, both matrices are 2x1. Therefore, they can indeed be multiplied under special rules due to their conformability in size.

Matrix Entry Multiplication

Once we’ve established that the matrices can be multiplied based on their dimensions, we move on to entry-wise multiplication. In this step, we multiply corresponding entries from each matrix.
For our case, we have:

• The first entry of the first matrix (4) multiplied by the first entry of the second matrix (5).
• The second entry of the first matrix (1) multiplied by the second entry of the second matrix (3).

This means that each element in the first matrix is multiplied by the element sitting in the same position in the second matrix. It's a simple multiplication of these entries without any additional steps at this point.

Matrix Summation

After performing the entry multiplication, the next step in matrix multiplication is to sum the products.
From our previous step, we have two products: 4 times 5 equals 20, and 1 times 3 equals 3. The last step is to add these products together.
The summation step is the final touch in matrix multiplication, where we combine all the products to get a single scalar value. For our exercise, adding 20 and 3 yields 23, which is the result of the multiplication. This summation solidifies your understanding and finalizes the operation we've been pursuing.