Problem 79 - Algebra and Trigonometry Real Mathematics, Real People

Question

Operations with Matrices Perform the operations (a) using a graphing utility and (b) by hand algebraically. If it is not possible to perform the operation(s), state the reason. $$\begin{aligned} &A=\left[\begin{array}{rrr} 1 & 2 & -2 \\ -1 & 1 & 0 \end{array}\right], \quad B=\left[\begin{array}{rrr} -1 & 4 & -1 \\ -2 & -1 & 0 \end{array}\right]\\\ &C=\left[\begin{array}{rr} 1 & 2 \\ -2 & 3 \\ 1 & 0 \end{array}\right], \quad c=-2, \text { and } d=-3 \end{aligned}$$ $$c d A+B$$

Step-by-Step Solution

Answer

The solution is \(cdA + B = 6A + B = [[6, 12, -12], [-6, 6, 0]] + [[-1, 4, -1], [-2, -1, 0]] = [[5, 16, -13], [-8, 5, 0]]\), where each entry is a result of scalar multiplication and matrix addition.

Step 1: Verify Operation Validity

First verify if the operations can be performed, given the dimensions of matrices \(A\), \(B\) and scalar quantities \(c\), \(d\). Both \(A\) and \(B\) are 2x3 matrices, which can be added together as they have same dimensions. The scalar multiplication \(cdA\) is possible since scalars can multiply any matrix.

Step 2: Perform Scalar Multiplication

Multiply the scalar quantities \(c\) and \(d\) together to get \(cd = -2 * -3 = 6\). Multiply this result with each entry of matrix \(A\). This will result in a new matrix \(cdA = 6A\), where every element of \(A\) is multiplied by 6.

Step 3: Perform Matrix Addition

Add together matrices \(cdA\) and \(B\). Do this by adding corresponding elements in the matrices to create a new matrix \(cdA + B\)

Key Concepts

Scalar Multiplication

When performing matrix operations, scalar multiplication is a fundamental step. It involves multiplying every entry of a matrix by a scalar, which is simply a constant number. For example, if you have a matrix \( A \) and a scalar \( c \), scalar multiplication results in a new matrix \( cA \) where each element of \( A \) is multiplied by \( c \). This operation scales all the values in the matrix, effectively stretching or compressing it by the factor of the scalar.

• Example: If \( c = 2 \) and \( A = \begin{bmatrix} 1 & 2 \ -1 & 0 \end{bmatrix} \), then \( cA = \begin{bmatrix} 2 \times 1 & 2 \times 2 \ 2 \times -1 & 2 \times 0 \end{bmatrix} = \begin{bmatrix} 2 & 4 \ -2 & 0 \end{bmatrix} \).
• It is commonly used in algebraic expressions involving matrices.

This operation is straightforward since it doesn't depend on the matrices' dimensions.

Matrix Addition

Matrix addition is the process of adding two matrices by adding their corresponding entries. For matrices to be added, they must be of the same dimensions, meaning they have the same number of rows and columns. This compatibility is crucial for matrix addition.

• Requirement: Matrices must have the same dimensions (e.g., both are 2x2 or 3x3).
• Result: The sum of two matrices yields another matrix with the same dimensions.

For instance, if \( A = \begin{bmatrix} 1 & 2 \ -1 & 0 \end{bmatrix} \) and \( B = \begin{bmatrix} 3 & 0 \ 4 & 1 \end{bmatrix} \), then \( A + B = \begin{bmatrix} 1+3 & 2+0 \ -1+4 & 0+1 \end{bmatrix} = \begin{bmatrix} 4 & 2 \ 3 & 1 \end{bmatrix} \). This step is essential to solve equations involving matrices.

Algebraic Operations

In algebraic operations with matrices, you often combine scalar multiplication and matrix addition, among other processes. These operations extend into solving matrices within equations, simplifying expressions, or even evaluating more complex algebraic terms.

• First, confirm that the operations are permissible by ensuring each action respects the rules of matrix dimensions.
• Complex algebraic expressions with matrices might require multiple operations in sequence—like in our exercise involving scalar multiplication followed by matrix addition.

For instance, combining multiple matrices entails meticulous execution. Each operation, such as multiplying scalars \( c \) and \( d \) to form \( cd \), precedes its application on matrix \( A \). It's a methodical approach ensuring each step adheres to algebraic principles, leading to a correct solution.

Graphing Utility

Graphing utilities are powerful tools that help visualize and perform matrix operations efficiently. These computational tools allow you to execute tasks like scalar multiplication, matrix addition, and more, providing quick and often instantaneous results. They are beneficial particularly in educational settings or when verifying your manually calculated results.

• Advantages: Saves time, reduces errors, and aids in visualizing complex operations.
• Uses: Ideal for graphically representing the results of matrix operations and verifying the manual work.

For example, using a graphing utility during matrix operations enables you to see the immediate effect of scalar multiplication or matrix addition. It's like having a calculator on steroids for matrices, ensuring that mathematical efforts are both visually engaging and incredibly accurate.