Problem 30 - College Algebra with Corequisite Support

Question

For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. (Hint: \(A^{2}=A \cdot A\) ) \(A=\left[\begin{array}{rr}-10 & 20 \\ 5 & 25\end{array}\right], B=\left[\begin{array}{rr}40 & 10 \\ -20 & 30\end{array}\right], C=\left[\begin{array}{rr}-1 & 0 \\ 0 & -1 \\ 1 & 0\end{array}\right]\) \(A B\)

Step-by-Step Solution

Answer

The product \(AB\) is \(\begin{bmatrix} -800 & 500 \\ -300 & 800 \end{bmatrix}\).

Step 1: Verify Matrix Dimensions

To perform matrix multiplication, we need to verify that the dimensions of the matrices are compatible. Matrix \(A\) is a \(2 \times 2\) matrix and matrix \(B\) is also a \(2 \times 2\) matrix. Because the number of columns in \(A\) is equal to the number of rows in \(B\), the multiplication is possible.

Step 2: Setup Matrix Multiplication Formula

Matrix multiplication of \(A\) and \(B\), i.e., \(AB\), is done by taking the dot product of rows from matrix \(A\) with columns from matrix \(B\). The resulting matrix will have dimensions \(2 \times 2\).

Step 3: Calculate Element (1,1)

The element at row 1, column 1 of the resulting matrix is calculated as: \((-10) \times 40 + 20 \times (-20) = -400 - 400 = -800\).

Step 4: Calculate Element (1,2)

The element at row 1, column 2 is calculated as: \((-10) \times 10 + 20 \times 30 = -100 + 600 = 500\).

Step 5: Calculate Element (2,1)

The element at row 2, column 1 is calculated as: \(5 \times 40 + 25 \times (-20) = 200 - 500 = -300\).

Step 6: Calculate Element (2,2)

The element at row 2, column 2 is calculated as: \(5 \times 10 + 25 \times 30 = 50 + 750 = 800\).

Step 7: Write the Resulting Matrix

After calculating each element, the resulting matrix \(AB\) is \(\begin{bmatrix} -800 & 500 \ -300 & 800 \end{bmatrix}\).

Key Concepts

Matrix Dimensions

When dealing with matrix multiplication, one of the first aspects to consider is matrix dimensions. The dimensions refer to the number of rows and columns that a matrix has. This is generally denoted as "rows × columns".
For example, a matrix with 2 rows and 3 columns would be a 2×3 matrix.
Understanding these dimensions is crucial because they determine whether two matrices can be multiplied together. The rule to remember is that the number of columns in the first matrix must match the number of rows in the second matrix.

• If matrix A is a 2×3 matrix and matrix B is a 3×4 matrix, then the matrices can be multiplied because the inner dimensions (3 and 3) are the same.
• The resulting product of multiplying these matrices will be a new matrix with dimensions from the outer sizes of the original matrices, in this case, a 2×4 matrix.
• If the dimensions don't align this way, the matrices cannot be multiplied.

In our example, both matrices A and B are 2×2 matrices, making them compatible for multiplication as their dimensions align perfectly.

Dot Product

The dot product is an essential part of matrix multiplication. To compute any single element in the resulting matrix, the dot product concept is used. This involves multiplying corresponding elements from a row of the first matrix and a column of the second matrix, then summing the results.
Let's walk through it step-by-step with two rows and columns from matrices A and B.

• First, take the row vector from matrix A. For example, \([-10, 20]\).
• Next, take the column vector from matrix B. For example, \(\begin{bmatrix} 40 \ -20 \end{bmatrix}\).
• Multiply the corresponding elements: \((-10 \times 40) = -400\) and \(20 \times (-20) = -400\).
• Add these products together to get the final element: \(-400 - 400 = -800\).

This calculated element \(-800\) then occupies the position in the resulting matrix that corresponds to the row from matrix A and the column from matrix B.

Matrix Notation

Matrix notation is the standardized format used to refer to and present matrices. It typically uses uppercase letters like A, B, or C to name matrices, and sometimes employs brackets to display the arrangement of elements within them.
Matrix notation clearly communicates important information about the matrix, including its elements and dimensions. Consider a sample matrix A:a = \[\begin{bmatrix}-10 & 20 \5 & 25\end{bmatrix}\]Here are some key components of matrix notation:

• The dimensions of the matrix are denoted as 2×2 because there are two rows and two columns.
• Each element within the matrix is usually represented by lowercase letters with subscripts indicating their position, such as \(a_{ij}\), where \(i\) is the row number and \(j\) is the column number.
• For example, \(a_{11} = -10\) and \(a_{22} = 25\).

Using this notation simplifies complex mathematical operations, allowing for succinct representation of the matrix's structure and facilitating clear communication when performing tasks like matrix multiplication.

Matrix Operations

Matrix operations include a variety of actions that can be performed on matrices, such as addition, subtraction, and multiplication, with each operation having specific rules and outcomes. Focusing on multiplication, as this operation can produce new matrices from given ones, offers insightful exploration.
For multiplication, the matrices must have compatible dimensions, as discussed earlier. If two matrices, A and B, meet the dimensionality requirement, you can proceed with the multiplication process.

• Identify the dimensions to ensure compatibility.
• Calculate each element in the resulting matrix by taking the dot product of appropriate rows and columns.
• As a result of multiplication, the new matrix will have dimensions equivalent to the number of rows of the first matrix and the number of columns of the second matrix.

In our exercise involving matrices A and B, the matrix operation of multiplication resulted in the matrix AB with elements \[ \begin{bmatrix} -800 & 500 \ -300 & 800 \end{bmatrix}\]. Practicing matrix operations helps you understand complex data interactions through linear transformations and dimensional analysis, applicable across various fields like computer graphics, physics, and economics.