Problem 67 - A Graphical Approach to Precalculus with Limits

Question

Find each matrix product if possible. $$\left[\begin{array}{ll}p & q \\ r & s\end{array}\right]\left[\begin{array}{ll}a & c \\ b & d\end{array}\right]$$

Step-by-Step Solution

Answer

The matrix product is \( \begin{pmatrix} pa + qb & pc + qd \\ ra + sb & rc + sd \end{pmatrix} \).

Step 1: Determine Matrix Dimensions

First, identify the dimensions of the matrices. The first matrix \( A \) is a 2x2 matrix: \( \begin{pmatrix} p & q \ r & s \end{pmatrix} \). The second matrix \( B \) is also a 2x2 matrix: \( \begin{pmatrix} a & c \ b & d \end{pmatrix} \). Both matrices are 2x2, so they can be multiplied together.

Step 2: Set Up Matrix Multiplication

To multiply two matrices, \( A \cdot B \), we calculate each element of the resulting matrix using the dot product of the corresponding row of the first matrix with the corresponding column of the second matrix. The resulting matrix will also be a 2x2 matrix.

Step 3: Calculate First Row of the Product Matrix

Compute the elements of the first row of the product matrix:\((1,1)\) entry: \( (pa + qb) = (p \cdot a) + (q \cdot b) \).\((1,2)\) entry: \( (pc + qd) = (p \cdot c) + (q \cdot d) \).

Step 4: Calculate Second Row of the Product Matrix

Compute the elements of the second row of the product matrix:\((2,1)\) entry: \( (ra + sb) = (r \cdot a) + (s \cdot b) \).\((2,2)\) entry: \( (rc + sd) = (r \cdot c) + (s \cdot d) \).

Step 5: Write the Resulting Matrix

The resulting product matrix is:\[ \begin{pmatrix} pa + qb & pc + qd \ ra + sb & rc + sd \end{pmatrix} \]

Key Concepts

Matrix Dimensions

Understanding matrix dimensions is crucial when working with matrix multiplication. Each matrix has a size described by the number of rows and columns it contains. This is noted as 'rows x columns'.
For example:

• A matrix with 3 rows and 4 columns has dimensions 3x4.
• A matrix with dimensions 2x3 has 2 rows and 3 columns.

In the context of multiplication, it's important to match the number of columns in the first matrix with the number of rows in the second matrix. If this condition fails, the matrices can't be multiplied. In our specific exercise, both matrices are 2x2, which means they each have 2 rows and 2 columns. Thus, they are compatible for multiplication.

Dot Product

The dot product plays a central role in computing the elements of a product matrix during matrix multiplication. Each element of the resulting matrix is obtained through the dot product of a row from the first matrix and a column from the second matrix.
To compute the dot product:

• Multiply the corresponding components of the row and the column.
• Add these products together.

For instance, to find the element in the first row and first column of a product matrix, we would multiply the elements of the first row from the first matrix by the corresponding elements of the first column from the second matrix, and add the results.

2x2 Matrix

A 2x2 matrix consists of two rows and two columns. These matrices are often used due to their simplicity and ease of manipulation in mathematical applications.
A 2x2 matrix can be represented as:\[\begin{pmatrix} a & b \c & d \end{pmatrix} \]To multiply two 2x2 matrices, the resulting matrix will also be a 2x2 matrix. Simple as they are, 2x2 matrices offer a good entry point for understanding bigger matrices. Moreover, each operation involves straightforward arithmetic.

Matrix Product

Matrix multiplication, often called the matrix product, involves creating a new matrix from two given matrices. Once you know the matrix dimensions and understand the dot product, performing matrix multiplication becomes quite procedural.
For our exercise, the matrices are:

• \[A = \begin{pmatrix} p & q \ r & s \end{pmatrix}\]
• \[B = \begin{pmatrix} a & c \ b & d \end{pmatrix}\]

The resulting matrix from their multiplication will have the form:\[\begin{pmatrix} pa + qb & pc + qd \ra + sb & rc + sd \end{pmatrix} \]This computation involves taking each element of the resulting matrix and filling it with values calculated using the dot product.