Problem 55 - Precalculus with Limits

Question

In Exercises 53-56, evaluate the expression. Use the matrix capabilities of a graphing utility to verify your answer. \(\left[\begin{array}{r} 0 && 2 & -2 \\ 4 && 1 & 2 \end{array}\right] \left(\left[\begin{array}{r} 4 & 0 \\ 0 & -1 \\ -1 & 2 \end{array}\right] + \left[\begin{array}{r} -2 & 3 \\ -3 & 5 \\ 0 & -3 \end{array}\right]\right)\)

Step-by-Step Solution

Answer

The result of the expression is \[ \left[\begin{array}{r} -3 && 0 \ 1 && 10 \end{array}\right] \]

Step 1: Add the Matrices

The two matrices to be added are \[ \left[\begin{array}{r} 4 & 0 \ 0 & -1 \ -1 & 2 \end{array}\right] \] and \[ \left[\begin{array}{r} -2 & 3 \ -3 & 5 \ 0 & -3 \end{array}\right] \]. To add two matrices, simply add the corresponding elements in each matrix. The result of the addition is \[ \left[\begin{array}{r} 2 & 3 \ -3 & 4 \ -1 & -1 \end{array}\right] \]

Step 2: Multiply with the Matrix

The matrix we are multiplying with is \[ \left[\begin{array}{r} 0 && 2 & -2 \ 4 && 1 & 2 \end{array}\right] \]. To perform the multiplication, we need to take each row of the first matrix, and do a dot product with each column of the second matrix. The result will be a new matrix with dimensions equal to the number of rows in the first matrix by the number of columns in the second matrix. The resulting matrix is \[ \left[\begin{array}{r} -3 && 0 \ 1 && 10 \end{array}\right] \]

Key Concepts

Matrix Addition

When tackling the concept of matrix addition, it's crucial to understand that matrices can only be added if they are of the same size. That means they must have the same number of rows and columns. In our exercise, the matrices being added are both 3x2, aligning perfectly for addition. To execute the addition, one simply adds the corresponding elements from each matrix together.

For example, to add the elements in the first row and first column of two matrices, you would calculate something like this: \( a_{11} + b_{11} \), where \( a_{11} \) and \( b_{11} \) are the elements in the first row and first column of matrix A and B, respectively. Applying this principle, the students easily found the sum of the two matrices in the problem, element by element.

Matrix Multiplication

Moving on to matrix multiplication, we enter a somewhat more complex territory. Unlike matrix addition, matrix multiplication doesn’t require matrices to have the same size. However, the number of columns in the first matrix must be equal to the number of rows in the second matrix.

The exercise involved multiplying a 2x3 matrix with a 3x2 matrix. The multiplication is done by taking the dot product of each row of the first matrix with each column of the second matrix to form a new element in the resulting matrix. This is done for all possible combinations of rows and columns, leading to a final matrix that has the same number of rows as the first matrix and the same number of columns as the second matrix.

Graphing Utility

A graphing utility is a powerful tool for students tackling matrix operations. It can perform fast calculations and visualizations which are invaluable, especially when dealing with larger matrices or complex operations. In precalculus, students are often encouraged to verify their answers using this technology.

Graphing utilities can instantly perform matrix addition and multiplication, returning results that can be compared against manual calculations. This can help in checking the correctness of one's work or in understanding where mistakes may have occurred. Students will interact with the utility, inputting matrices and observing results, which reinforces their learning and comprehension of matrix operations.

Dot Product

The dot product is a key operation within matrix multiplication. It involves multiplying corresponding entries and then summing those products. When we multiply a row from one matrix by a column from another, we are essentially performing a dot product.

In the given problem, for instance, to multiply the 2x3 matrix by the 3x2 matrix, each element of the resulting 2x2 matrix is derived from a dot product. The students learned to pair the first row with the first column of the other matrix, multiply the corresponding elements, sum them up, and place the result in the appropriate position in the new matrix. This operation is the building block of matrix multiplication and requires careful attention to detail to ensure accuracy in calculations.