Problem 59 - A Graphical Approach to College Algebra

Question

Find each matrix product if possible. $$\left[\begin{array}{rr} 5 & 2 \\ -1 & 4 \end{array}\right]\left[\begin{array}{rr} 3 & -2 \\ 1 & 0 \end{array}\right]$$

Step-by-Step Solution

Answer

The matrix product is \( \begin{bmatrix} 17 & -10 \\ 1 & 2 \end{bmatrix} \).

Step 1: Confirm Matrix Dimensions

The first matrix has dimensions 2x2 and the second matrix also has dimensions 2x2. Since the number of columns in the first matrix equals the number of rows in the second matrix, the matrices can be multiplied.

Step 2: Set Up the Product

Let's denote the matrices as \( A \) and \( B \):\[ A = \begin{bmatrix} 5 & 2 \ -1 & 4 \end{bmatrix}, B = \begin{bmatrix} 3 & -2 \ 1 & 0 \end{bmatrix} \]The resultant matrix will also have dimensions 2x2. Let's denote it as \( C \), where:\[ C = A \cdot B = \begin{bmatrix} c_{11} & c_{12} \ c_{21} & c_{22} \end{bmatrix} \]

Step 3: Calculate the Entry \( c_{11} \)

To find \( c_{11} \), multiply the first row of \( A \) by the first column of \( B \):\[ c_{11} = (5 \times 3) + (2 \times 1) = 15 + 2 = 17 \]

Step 4: Calculate the Entry \( c_{12} \)

To find \( c_{12} \), multiply the first row of \( A \) by the second column of \( B \):\[ c_{12} = (5 \times -2) + (2 \times 0) = -10 + 0 = -10 \]

Step 5: Calculate the Entry \( c_{21} \)

To find \( c_{21} \), multiply the second row of \( A \) by the first column of \( B \):\[ c_{21} = (-1 \times 3) + (4 \times 1) = -3 + 4 = 1 \]

Step 6: Calculate the Entry \( c_{22} \)

To find \( c_{22} \), multiply the second row of \( A \) by the second column of \( B \):\[ c_{22} = (-1 \times -2) + (4 \times 0) = 2 + 0 = 2 \]

Step 7: Write the Resultant Matrix

The final matrix after performing all the calculations is:\[ C = \begin{bmatrix} 17 & -10 \ 1 & 2 \end{bmatrix} \]

Key Concepts

Matrix Dimensions

Matrix dimensions are crucial when dealing with matrix multiplication. Each matrix is defined by its rows and columns and are given in the format "rows x columns". This is important because in matrix multiplication, you need to check if the multiplication can actually be performed.
In general, for two matrices to be multiplied, the number of columns in the first matrix must match the number of rows in the second matrix.
Let's look at the example provided:

• The first matrix, often given as \( A \), is a \( 2 \times 2 \) matrix.
• The second matrix, \( B \), is also a \( 2 \times 2 \) matrix.
• Since the number of columns in \( A \) (which is 2) is equal to the number of rows in \( B \) (which is also 2), the multiplication is valid.

Knowing the dimensions tells you not only if multiplication is possible, but also the size of the resulting product matrix, which will have the number of rows of the first matrix and the number of columns of the second matrix.
Here, the product will also be a \( 2 \times 2 \) matrix.

Matrix Product

The matrix product itself is the result of performing matrix multiplication. It combines the information from both matrices into a single, new matrix. The calculation of this product might seem complicated at first, but it follows a straightforward process.
In our example, we define the matrices:

• \( A = \begin{bmatrix} 5 & 2 \ -1 & 4 \end{bmatrix} \)
• \( B = \begin{bmatrix} 3 & -2 \ 1 & 0 \end{bmatrix} \)

The resulting product matrix, \( C \), is calculated through the element-wise multiplication and sum of rows of \( A \) and columns of \( B \).
This operation combines corresponding elements and creates a new matrix:

• \( C = A \cdot B = \begin{bmatrix} c_{11} & c_{12} \ c_{21} & c_{22} \end{bmatrix} \)

It’s important to note that the order matters—the matrix \( A \cdot B \) is not generally the same as \( B \cdot A \). Matrix multiplication is not commutative.

Matrix Entry Calculation

Each entry in the product matrix is calculated by taking the dot product of the corresponding row from the first matrix and the column from the second matrix. This involves multiplying each pair of corresponding entries and adding them together.
Let's look at the calculations for the entries in the matrix \( C \).

Computing the Entries

• For \( c_{11} \): Multiply the first row of \( A \) by the first column of \( B \):
  \( c_{11} = (5 \times 3) + (2 \times 1) = 15 + 2 = 17 \)
• For \( c_{12} \): Multiply the first row of \( A \) by the second column of \( B \):
  \( c_{12} = (5 \times -2) + (2 \times 0) = -10 + 0 = -10 \)
• For \( c_{21} \): Multiply the second row of \( A \) by the first column of \( B \):
  \( c_{21} = (-1 \times 3) + (4 \times 1) = -3 + 4 = 1 \)
• For \( c_{22} \): Multiply the second row of \( A \) by the second column of \( B \):
  \( c_{22} = (-1 \times -2) + (4 \times 0) = 2 + 0 = 2 \)

Each of these calculations results in a value that contributes to the final product matrix, \( C = \begin{bmatrix} 17 & -10 \ 1 & 2 \end{bmatrix} \).
It's a systematic process for finding each entry step by step.