Problem 54 - Precalculus with Limits

Question

In Exercises 53-56, evaluate the expression. Use the matrix capabilities of a graphing utility to verify your answer. \(-3\left(\left[\begin{array}{r} 6 & 5 & -1 \\ 1 & -2 & 0 \end{array}\right]\left[\begin{array}{r} 0 & 3 \\ -1 & -3 \\ 4 & 1 \end{array}\right] \right)\)

Step-by-Step Solution

Answer

The result of the exercise is the matrix \(\left[\begin{array}{r} -3 & -27 \ 6 & 15 \end{array}\right]\)

Step 1: Matrix Multiplication

Determine the matrix multiplication of \(\left[\begin{array}{r} 6 & 5 & -1 \ 1 & -2 & 0\end{array}\right]\) and \(\left[\begin{array}{r} 0 & 3 \ -1 & -3 \ 4 & 1\end{array}\right]\). Start by multiplying the elements of the first row of the first matrix with the elements of the first column of the second matrix, add those products, this gives you the first element of the new matrix. Repeat the process for the rest of the elements.

Step 2: Matrix Multiplication Result

The result of the matrix multiplication is \(\left[\begin{array}{r} 1 & 9 \ -2 & -5 \end{array}\right]\)

Step 3: Scalar Multiplication

Now, multiply this result by the scalar -3. This is accomplished by multiplying each term of the matrix by -3.

Step 4: Scalar Multiplication Result

The result of the scalar multiplication is \(\left[\begin{array}{r} -3 & -27 \ 6 & 15 \end{array}\right]\)

Key Concepts

Matrix Scalar Multiplication

Matrix scalar multiplication is the process of multiplying each element of a matrix by a single number called a scalar. This is a simple yet essential operation in linear algebra. If you have a matrix and a scalar, you apply the scalar to every element by multiplying it.
For instance, if we consider the matrix from our example, \[\left[ \begin{array}{rr} 1 & 9 \ -2 & -5 \end{array}\right] \]and the scalar -3, you would multiply each element by -3:

• 1 becomes -3
• 9 becomes -27
• -2 becomes 6
• -5 becomes 15

This yields a new matrix:\[\left[\begin{array}{rr}-3 & -27\ 6 & 15\end{array}\right]\]Simple, right? This process is useful because it scales or adjusts the entire matrix by the same factor, which is helpful in many applications such as stretching or shrinking geometrical figures, adjusting weights in data transformation, and more.

Graphing Utility Verification

Verifying results using a graphing utility is an excellent way to ensure accuracy when performing complex calculations by hand. Graphing calculators or similar software tools can carry out operations such as matrix multiplication and scalar multiplication quickly and correctly.
To verify the matrix product \[\left[\begin{array}{c}6 & 5 & -1\ 1 & -2 & 0\end{array}\right] \left[\begin{array}{c}0 & 3\ -1 & -3\ 4 & 1\end{array}\right]\]using a graphing utility:

• Input the matrices into the graphing utility.
• Use the matrix function to perform the multiplication.
• Check that the output matches the calculated result before scalar multiplication: \[\left[\begin{array}{rr}1 & 9\ -2 & -5\end{array}\right]\]
• Then verify scalar multiplication by multiplying the output by -3.

This will help in catching errors or confirming your manual calculation. It's a reliable method to build confidence in your calculations.

Matrix Operations

Matrix operations are the cornerstone of linear algebra, combining matrices and operations such as addition, subtraction, multiplication, and scalar multiplication. Each operation has its own set of rules and procedures.
In our exercise, we focus on matrix multiplication followed by scalar multiplication:

• Matrix Multiplication: This involves taking the row elements from the first matrix and multiplying them against the column elements of the second matrix, then each product is summed up to produce an element for a new matrix. This process continues across all rows and columns.
• Scalar Multiplication: Once the new matrix is created, multiplying each of its elements by an external number or scalar fine-tunes the matrix for further use or interpretation.

These operations are fundamental because they form the basis for manipulating and transforming datasets in computer graphics, engineering, physics, and more. Understanding how they work will enhance your ability to solve practical problems involving systems of equations, transformations, and other analytical tasks.