Problem 52 - Algebra and Trigonometry Real Mathematics, Real People

Question

Operations with Matrices Use the matrix capabilities of a graphing utility to evaluate the expression. $$\left[\begin{array}{rrr} 6 & 5 & -1 \\ 1 & -2 & 0 \end{array}\right]\left[\begin{array}{rr} 0 & 3 \\ -1 & -3 \\ 4 & 1 \end{array}\right]\left[\begin{array}{rr} -2 & 2 \\ 0 & -1 \end{array}\right]$$

Step-by-Step Solution

Answer

The result of the matrix multiplication is \(\left[\begin{array}{rr}-10 & 1\-4 & 1\end{array}\right]\).

Step 1: Multiply the First Two Matrices

Begin by multiplying the first two matrices. This is accomplished by multiplying each element of a row in the first matrix by the respective element of the corresponding column in the second matrix, and then adding those products. The resultant matrix from multiplying \(\left[\begin{array}{rrr}6 & 5 & -1\1 & -2 & 0\end{array}\right]\) and \(\left[\begin{array}{rr}0 & 3\-1 & -3\4 & 1\end{array}\right]\) is \(\left[\begin{array}{rr}5 & 9\-2 & 3\end{array}\right]\).

Step 2: Multiply the Resultant Matrix with the Third Matrix

Now, multiply this resultant matrix with the third matrix. Using the same process, you will find that the resultant matrix from multiplying \(\left[\begin{array}{rr}5 & 9\-2 & 3\end{array}\right]\) and \(\left[\begin{array}{rr}-2 & 2\0 & -1\end{array}\right]\) is \(\left[\begin{array}{rr}-10 & 1\-4 & 1\end{array}\right]\).

Key Concepts

Operations with matrices

Matrix multiplication is an essential concept in algebra that represents an array of numbers, symbols, or expressions arranged in rows and columns. To perform matrix multiplication, certain rules must be followed.
The first rule is that the number of columns in the first matrix must be equal to the number of rows in the second matrix. The product of a matrix multiplication will result in a new matrix where the number of rows is determined by the first matrix and the number of columns by the second matrix.

The step-by-step multiplication involves taking a row from the first matrix and a column from the second matrix. You then multiply each corresponding element and sum up all these products to get a single entry in the result matrix. Repeat this process for each row of the first matrix, and for each column of the second matrix, until all entries are calculated.

• Consider the elements in the row of the first matrix.
• Multiply them with the corresponding elements in the column of the second matrix.
• Add up these products to obtain a single number.
• Place this number in the corresponding position in the result matrix.
• Continue this operation for each row and column.

Understanding these operations is critical, as matrix multiplication is used in various applications, including in computer graphics, coding theory, and statistics.

Graphing Utility

A graphing utility is a powerful tool often used in mathematics, particularly when working with matrices. It can simplify complex calculations and provide visual insight into the data.

In our exercise, using a graphing utility would allow us to quickly compute the product of matrices without manual calculations, which can be susceptible to error, especially with larger matrices. The utility automates the multiplication process and usually provides a step-by-step display of the computation.

Utilizing a Graphing Utility:

Here's how you might use a graphing calculator to solve our matrix multiplication example:

• Input the first matrix into the calculator.
• Then, input the second matrix.
• Use the appropriate command to multiply the matrices.
• Repeat the process with the resulting matrix and the third matrix in the sequence.

Graphing utilities don't just calculate the entries of the product matrix; they often include features for performing a variety of other matrix operations such as determinant calculation, matrix inversion, and solving systems of linear equations. Learning to use these functions is invaluable for students tackling linear algebra or related subjects.

Elementary Row Operations

Elementary row operations are at the heart of many calculations with matrices. These operations consist of three fundamental manipulations: switching rows, multiplying a row by a non-zero constant, and adding or subtracting multiples of rows from one another.

These manipulations are essential for various processes, like solving systems of equations through methods such as Gaussian elimination, determining the rank of a matrix, and finding the inverse of a matrix, if it exists.

Roles in Matrix Multiplication:

Although not directly used in our exercise, understanding elementary row operations can greatly enhance one's grasp on manipulating matrices.

• Switching rows: This changes the order of the equations in a system but doesn't affect the solution.
• Scalar multiplication: Multiplying a row by a non-zero constant can help simplify equations or matrix entries.
• Row addition or subtraction: This can help eliminate variables in a system of equations or simplify a matrix during operations like finding its inverse.

Having a firm handle on these operations not only aids in matrix multiplication but also in broader algebraic contexts within mathematics.