Problem 26 - A Graphical Approach to College Algebra

Question

Perform each operation if possible. $$\left[\begin{array}{rrr} 6 & -2 & 4 \\ -2 & 5 & 8 \\ 1 & 0 & 2 \end{array}\right]+\left[\begin{array}{rrr} 3 & 0 & 8 \\ 1 & -2 & 4 \\ 6 & 9 & -2 \end{array}\right]-\left[\begin{array}{rrr} -4 & 2 & 1 \\ 0 & 3 & -2 \\ 4 & 2 & 0 \end{array}\right]$$

Step-by-Step Solution

Answer

The final result is \(\begin{bmatrix} 13 & -4 & 11 \\ -1 & 0 & 14 \\ 3 & 7 & 0 \end{bmatrix}\).

Step 1: Verify Matrix Dimensions

Before performing any operations, we must ensure that all matrices have the same dimensions. Each matrix has 3 rows and 3 columns, so all operations are valid.

Step 2: Perform the Addition

Add the corresponding elements of the first and second matrices:\[\begin{bmatrix}6+3 & -2+0 & 4+8 \-2+1 & 5+(-2) & 8+4 \1+6 & 0+9 & 2+(-2)\end{bmatrix}=\begin{bmatrix}9 & -2 & 12 \-1 & 3 & 12 \7 & 9 & 0\end{bmatrix}\]

Step 3: Perform the Subtraction

Subtract the corresponding elements of the third matrix from the resulting matrix of the addition:\[\begin{bmatrix}9-(-4) & -2-2 & 12-1 \-1-0 & 3-3 & 12-(-2) \7-4 & 9-2 & 0-0\end{bmatrix}=\begin{bmatrix}13 & -4 & 11 \-1 & 0 & 14 \3 & 7 & 0\end{bmatrix}\]

Step 4: Write the Final Result

The result of the operation \(\begin{bmatrix} A \end{bmatrix} + \begin{bmatrix} B \end{bmatrix} - \begin{bmatrix} C \end{bmatrix}\) is:\[\begin{bmatrix} 13 & -4 & 11 \ -1 & 0 & 14 \ 3 & 7 & 0 \end{bmatrix}\]

Key Concepts

Matrix Addition

Matrix addition is the process of adding two matrices by adding the corresponding elements together. For two matrices to be added, they must have the same dimensions. This means they should have the same number of rows and columns. In our exercise, each matrix is a 3x3 matrix, allowing for a straightforward addition process.

• Start by adding the elements in the first row of both matrices. For example: in the first column of both matrices, you will add 6 from the first matrix to 3 from the second, giving a result of 9.
• Continue this process for each corresponding element: - Second column, first row: - Second column, second row: -2 from the first matrix added with 0 from the second becomes -2.

The rule is simple: each element is added to its corresponding element. This will result in a new matrix where each element represents the sum of the corresponding elements from the original matrices. The final result of this addition in our exercise is the matrix \[\begin{bmatrix} 9 & -2 & 12 \ -1 & 3 & 12 \ 7 & 9 & 0 \end{bmatrix}\]

Matrix Subtraction

Matrix subtraction is similar to matrix addition but involves subtraction of corresponding elements rather than addition. Just like addition, the matrices need to have the same dimensions to perform matrix subtraction.

• Consider the two matrices we obtained after the addition step and the third matrix from the original exercise.
• Subtract each element in the corresponding position of the third matrix from the resulting matrix of the addition.
• For example:
  • The first element of the first row after subtraction is calculated by taking 9 minus (-4): (9 - (-4)) = 13.
  • In the same way, subtract corresponding elements in other positions: - Second column, first row: -2 minus 2 equals -4.

Each element in the resultant matrix is created by a simple element-wise subtraction between the matrices. This gives us the final matrix \[\begin{bmatrix} 13 & -4 & 11 \ -1 & 0 & 14 \ 3 & 7 & 0 \end{bmatrix}\]

Matrices

A matrix is a rectangular array of numbers arranged in rows and columns. Each number in a matrix is an element. Matrices are essential in various scientific fields such as mathematics, computer science, and engineering because they can represent and solve linear equations.

• Look at our example matrices: \[\begin{bmatrix} 6 & -2 & 4 \ -2 & 5 & 8 \ 1 & 0 & 2 \end{bmatrix}\]Here, each entry is an integer, and the entire structure has 3 rows and 3 columns, making it a 3x3 matrix.
• Dimensions: Matrices must have compatible dimensions to perform operations like addition and subtraction. In our exercise, all matrices have dimensions of 3x3, making all operations possible.
• Use of Matrices:
  • Within mathematics, matrices are used to solve linear equations. This is often seen in physics and engineering problems.
  • Besides operations like addition and subtraction, matrices can be used in more complex operations such as multiplication, determinant computation, and finding inverses.

Matrices provide a structure that can be easily manipulated with basic arithmetic to solve numerous problems across various disciplines.