Problem 5 - Precalculus: Functions and Graphs

Question

Find, if possible, \(A+B, A-B, 2 A,\) and \(-3 B\). $$A=\left[\begin{array}{lll} 4 & -3 & 2 \end{array}\right], \quad B=\left[\begin{array}{lll} 7 & 0 & -5 \end{array}\right]$$

Step-by-Step Solution

Answer

\( A+B = \begin{bmatrix} 11 & -3 & -3 \end{bmatrix}, A-B = \begin{bmatrix} -3 & -3 & 7 \end{bmatrix}, 2A = \begin{bmatrix} 8 & -6 & 4 \end{bmatrix}, -3B = \begin{bmatrix} -21 & 0 & 15 \end{bmatrix} \)

Step 1: Verify Dimensions for Addition and Subtraction

The matrix addition and subtraction require matrices to have the same dimensions. Matrix \(A\) is a \(1 \times 3\) row matrix, and matrix \(B\) is also a \(1 \times 3\) row matrix. Thus, their dimensions match, and we can perform addition and subtraction.

Step 2: Matrix Addition \(A + B\)

To add matrices \(A\) and \(B\), add corresponding elements:\[A + B = \begin{bmatrix} 4 & -3 & 2 \end{bmatrix} + \begin{bmatrix} 7 & 0 & -5 \end{bmatrix} = \begin{bmatrix} 4+7 & -3+0 & 2+(-5) \end{bmatrix} = \begin{bmatrix} 11 & -3 & -3 \end{bmatrix}\]

Step 3: Matrix Subtraction \(A - B\)

Subtract corresponding elements of \(B\) from \(A\):\[A - B = \begin{bmatrix} 4 & -3 & 2 \end{bmatrix} - \begin{bmatrix} 7 & 0 & -5 \end{bmatrix} = \begin{bmatrix} 4-7 & -3-0 & 2-(-5) \end{bmatrix} = \begin{bmatrix} -3 & -3 & 7 \end{bmatrix}\]

Step 4: Scalar Multiplication \(2A\)

Multiply each element of \(A\) by 2:\[2A = 2 \times \begin{bmatrix} 4 & -3 & 2 \end{bmatrix} = \begin{bmatrix} 2 \times 4 & 2 \times (-3) & 2 \times 2 \end{bmatrix} = \begin{bmatrix} 8 & -6 & 4 \end{bmatrix}\]

Step 5: Scalar Multiplication \(-3B\)

Multiply each element of \(B\) by -3:\[-3B = -3 \times \begin{bmatrix} 7 & 0 & -5 \end{bmatrix} = \begin{bmatrix} -3 \times 7 & -3 \times 0 & -3 \times (-5) \end{bmatrix} = \begin{bmatrix} -21 & 0 & 15 \end{bmatrix}\]

Key Concepts

Matrix Addition

Matrix addition is a simple yet fundamental operation in the realm of matrices. To add two matrices together, it is crucial that both matrices share the same dimensions. This means the matrices must have the same number of rows and columns.
In the case of our exercise, matrices \(A\) and \(B\) both share the dimensions of a \(1 \times 3\) row matrix. Thus, matrix addition can proceed without any issues.
To perform the addition \(A + B\), you simply add the corresponding elements from each matrix. Let's break it down:

• The first element in row 1 of \(A\) is added to the first element in row 1 of \(B\): \(4 + 7 = 11\).
• Do the same for the second and third elements: \(-3 + 0 = -3\) and \(2 + (-5) = -3\).

Therefore, the result of the matrix addition is the new row matrix \(\begin{bmatrix} 11 & -3 & -3 \end{bmatrix}\). Each element in the resulting matrix is the sum of the corresponding elements of \(A\) and \(B\).

Matrix Subtraction

Matrix subtraction is very similar to matrix addition. Like addition, subtraction of two matrices is only possible if they have the same dimensions. Fortunately, matrices \(A\) and \(B\) are both row matrices with the dimensions \(1 \times 3\).
To find the result of \(A - B\), subtract corresponding elements in matrix \(B\) from those in matrix \(A\). Break it down into parts:

• Subtract the first element of \(B\) from \(A\): \(4 - 7 = -3\).
• Repeat for the second and third elements: \(-3 - 0 = -3\) and \(2 - (-5) = 7\).

The resulting matrix after subtraction is \(\begin{bmatrix} -3 & -3 & 7 \end{bmatrix}\). Each element represents the difference of corresponding elements from matrices \(A\) and \(B\).

Scalar Multiplication

Scalar multiplication involves multiplying every entry of a matrix by the same scalar value. It essentially scales the matrix up or down without changing its structure. In our exercise, two types of scalar multiplication are demonstrated: multiplying \(A\) by 2 and \(B\) by -3.
For \(2A\), each element in matrix \(A\) is multiplied by 2:

• The first element: \(2 \times 4 = 8\).
• The second element: \(2 \times -3 = -6\).
• The third element: \(2 \times 2 = 4\).

This results in the matrix \(\begin{bmatrix} 8 & -6 & 4 \end{bmatrix}\).
Similarly, when calculating \(-3B\), each element in matrix \(B\) is multiplied by -3:

• The first element: \(-3 \times 7 = -21\).
• The second element: \(-3 \times 0 = 0\).
• The third element: \(-3 \times -5 = 15\).

The resulting matrix is \(\begin{bmatrix} -21 & 0 & 15 \end{bmatrix}\). Scalar multiplication is helpful for changing the magnitude of matrices while keeping their relative values consistent.

Row Matrix

A row matrix is a single row of elements, which can contain one or more columns. In mathematics and linear algebra, a row matrix is considered a special type of matrix as its structure is linear with only one row.
Matrices \(A\) and \(B\) in our exercise are examples of row matrices because they consist of just one row each. Specifically, both are \(1 \times 3\) matrices, meaning they have one row and three columns.
Row matrices are often used in various mathematical computations because they represent linear data configurations. They are particularly useful in:

• Vector representation, where a row matrix can act as a vector in linear algebra contexts.
• Representing linear combinations and transformations, as they can be easily combined or manipulated through operations like addition and scalar multiplication.

Due to their simplicity, row matrices are excellent for visualizing basic matrix operations, such as those illustrated by our exercise. Understanding row matrices and how they interact with other matrices is essential for mastering matrix algebra.