Problem 29 - A Graphical Approach to College Algebra

Question

Perform each operation if possible. $$2\left[\begin{array}{rrr} 2 & -1 & -1 \\ -1 & 2 & -1 \\ -1 & -1 & 2 \end{array}\right]+3\left[\begin{array}{lll} 1 & 2 & 3 \\ 2 & 1 & 3 \\ 2 & 3 & 1 \end{array}\right]$$

Step-by-Step Solution

Answer

The resultant matrix is \(\begin{bmatrix} 7 & 4 & 7 \\ 4 & 7 & 7 \\ 4 & 7 & 7 \end{bmatrix}\).

Step 1: Understand the Scalars and Matrices

We have two matrices and two scalar multipliers. The matrices are of order 3x3, meaning they each have three rows and three columns. The first matrix will be multiplied by 2, and the second matrix by 3.

Step 2: Multiply the First Matrix by 2

Apply the scalar multiplication to \[\begin{bmatrix} 2 & -1 & -1 \ -1 & 2 & -1 \ -1 & -1 & 2 \end{bmatrix}\].Multiply each element by 2:\[\begin{bmatrix} 4 & -2 & -2 \ -2 & 4 & -2 \ -2 & -2 & 4 \end{bmatrix}\].

Step 3: Multiply the Second Matrix by 3

Apply the scalar multiplication to \[\begin{bmatrix} 1 & 2 & 3 \ 2 & 1 & 3 \ 2 & 3 & 1 \end{bmatrix}\].Multiply each element by 3:\[\begin{bmatrix} 3 & 6 & 9 \ 6 & 3 & 9 \ 6 & 9 & 3 \end{bmatrix}\].

Step 4: Add the Resultant Matrices

Now, add the matrices obtained from the previous steps:\[\begin{bmatrix} 4 & -2 & -2 \ -2 & 4 & -2 \ -2 & -2 & 4 \end{bmatrix}+\begin{bmatrix} 3 & 6 & 9 \ 6 & 3 & 9 \ 6 & 9 & 3 \end{bmatrix}\].Perform element-wise addition:\[\begin{bmatrix} 4+3 & -2+6 & -2+9 \ -2+6 & 4+3 & -2+9 \ -2+6 & -2+9 & 4+3 \end{bmatrix} =\begin{bmatrix} 7 & 4 & 7 \ 4 & 7 & 7 \ 4 & 7 & 7 \end{bmatrix}\].

Step 5: Verify the Result

Ensure that each corresponding element from the resultant matrices in Step 4 was correctly added. The final matrix is:\[\begin{bmatrix} 7 & 4 & 7 \ 4 & 7 & 7 \ 4 & 7 & 7 \end{bmatrix}\].

Key Concepts

Scalar Multiplication

Scalar multiplication is one of the foundational operations in matrix algebra. It involves multiplying every element of a matrix by a single number, known as a "scalar." This operation scales the matrix while maintaining its original structure.

For example, suppose you have a matrix \( A \) and a scalar \( k \). To perform scalar multiplication, you multiply each element of the matrix by \( k \). Here's how it works:

• Every element of the matrix \( A \) becomes \( k \times a_{ij} \), where \( a_{ij} \) is an element in the \( i \)-th row and \( j \)-th column.

In the original exercise, the first matrix was multiplied by 2. Each element like 2 becomes 4 after being multiplied by 2. This operation simplifies the matrix but keeps its order and structure.

Scalar multiplication is especially useful for transforming matrices, scaling them, or preparing them for further operations like matrix addition.

Matrix Addition

After performing scalar multiplication on each matrix, the next step involves matrix addition. Like adding numbers, matrix addition combines two matrices into one by adding their corresponding elements together.

To add two matrices:

• They must be of the same size. This means they have the same number of rows and columns.
• Each element of one matrix is added to the corresponding element of the other matrix.

The matrices derived from our scalar multiplication example were both 3x3 matrices. This allows us to add them together without any problem, ensuring each corresponding element is added to form a new resultant matrix. The result is a new matrix where each element is the sum of the elements from the two input matrices.

This process reinforces understanding of element-wise operations and the need for matrices to have the same dimensions in operations like addition.

Elementary Linear Algebra

Elementary linear algebra is a branch of mathematics concerning vector spaces and linear mappings between these spaces. It covers various operations, including the ones we've discussed: scalar multiplication and matrix addition.

Understanding these operations is crucial as they form the building blocks of more complex algebraic processes. They help solve systems of linear equations, manage transformations in vector spaces, and deal with real-world data within multidimensional arrays.

Exploring Concepts:

• **Matrix Operations:** These are techniques applied to matrices such as multiplication, addition, and inversion, often used in computational applications.
• **Linear Transformations:** These represent functions that map from one vector space to another while preserving vector addition and scalar multiplication.

Elementary linear algebra provides a toolkit that is applicable in various fields, including computer science, physics, and engineering. By understanding how to manipulate matrices through operations like scalar multiplication and addition, students can better apply these concepts to real-world problems.