Problem 6 - Algebra for College Students

Question

For Problems \(1-8\), find \(A+B, A-B, 2 A+3 B\), and \(4 A-2 B\). $$ A=\left[\begin{array}{rr} 7 & -4 \\ -5 & 9 \\ -1 & 2 \end{array}\right], \quad B=\left[\begin{array}{rr} 12 & 3 \\ -2 & -4 \\ -6 & 7 \end{array}\right] $$

Step-by-Step Solution

Answer

\(A+B = \begin{bmatrix} 19 & -1 \\ -7 & 5 \\ -7 & 9 \end{bmatrix}\); \(A-B = \begin{bmatrix} -5 & -7 \\ -3 & 13 \\ 5 & -5 \end{bmatrix}\); \(2A+3B = \begin{bmatrix} 50 & 1 \\ -16 & 6 \\ -20 & 25 \end{bmatrix}\); \(4A-2B = \begin{bmatrix} 4 & -22 \\ -16 & 44 \\ 8 & -6 \end{bmatrix}\).

Step 1: Verify Matrix Dimensions

First, check whether the matrices \(A\) and \(B\) have the same dimensions. Matrix \(A\) is a \(3 \times 2\) matrix and matrix \(B\) is also a \(3 \times 2\) matrix. Therefore, they can be added and subtracted.

Step 2: Find \(A + B\)

To add the matrices \(A\) and \(B\), add corresponding elements from each matrix:\[A + B = \begin{bmatrix} 7+12 & -4+3 \ -5-2 & 9-4 \ -1-6 & 2+7 \end{bmatrix} = \begin{bmatrix} 19 & -1 \ -7 & 5 \ -7 & 9 \end{bmatrix}\]

Step 3: Find \(A - B\)

To subtract matrix \(B\) from \(A\), subtract corresponding elements in \(B\) from \(A\):\[A - B = \begin{bmatrix} 7-12 & -4-3 \ -5+2 & 9+4 \ -1+6 & 2-7 \end{bmatrix} = \begin{bmatrix} -5 & -7 \ -3 & 13 \ 5 & -5 \end{bmatrix}\]

Step 4: Find \(2A + 3B\)

First, multiply each element of matrix \(A\) by 2:\[2A = \begin{bmatrix} 14 & -8 \ -10 & 18 \ -2 & 4 \end{bmatrix}\]Next, multiply each element of matrix \(B\) by 3:\[3B = \begin{bmatrix} 36 & 9 \ -6 & -12 \ -18 & 21 \end{bmatrix}\]Now, add \(2A\) and \(3B\):\[2A + 3B = \begin{bmatrix} 14+36 & -8+9 \ -10-6 & 18-12 \ -2-18 & 4+21 \end{bmatrix} = \begin{bmatrix} 50 & 1 \ -16 & 6 \ -20 & 25 \end{bmatrix}\]

Step 5: Find \(4A - 2B\)

First, multiply each element of matrix \(A\) by 4:\[4A = \begin{bmatrix} 28 & -16 \ -20 & 36 \ -4 & 8 \end{bmatrix}\]Next, multiply each element of matrix \(B\) by 2:\[2B = \begin{bmatrix} 24 & 6 \ -4 & -8 \ -12 & 14 \end{bmatrix}\]Now, subtract \(2B\) from \(4A\):\[4A - 2B = \begin{bmatrix} 28-24 & -16-6 \ -20+4 & 36+8 \ -4+12 & 8-14 \end{bmatrix} = \begin{bmatrix} 4 & -22 \ -16 & 44 \ 8 & -6 \end{bmatrix}\]

Key Concepts

Matrix Addition

Matrix addition is a simple yet crucial operation in linear algebra. It involves adding corresponding elements of two matrices, meaning each element in the resulting matrix is the sum of elements at the same position in the given matrices. For matrices to be added, they must have the same dimensions. In other words, if matrix \(A\) is a \(3 \times 2\) matrix, matrix \(B\) must also be a \(3 \times 2\) matrix for \(A + B\) to be defined.

To illustrate, consider the matrices:

• Matrix \(A = \begin{bmatrix} 7 & -4 \ -5 & 9 \ -1 & 2 \end{bmatrix}\)
• Matrix \(B = \begin{bmatrix} 12 & 3 \ -2 & -4 \ -6 & 7 \end{bmatrix}\)

To find \(A + B\), add each element:
- The first element in \(A\) (7) is added to the first element in \(B\) (12), resulting in 19.
- Continue this process for each pair of corresponding elements.

This operation gives the resulting matrix:\[A + B = \begin{bmatrix} 19 & -1 \ -7 & 5 \ -7 & 9 \end{bmatrix}\] which enables us to combine data from two similar matrices systematically.

Matrix Subtraction

Matrix subtraction is the process of subtracting corresponding elements of one matrix from another. Like addition, subtraction requires that both matrices have the same dimensions, ensuring that every element in the first matrix has a counterpart in the second matrix.

In our example, to calculate \(A - B\), we take the corresponding elements of matrices \(A\) and \(B\) and find the difference:

• Matrix \(A = \begin{bmatrix} 7 & -4 \ -5 & 9 \ -1 & 2 \end{bmatrix}\)
• Matrix \(B = \begin{bmatrix} 12 & 3 \ -2 & -4 \ -6 & 7 \end{bmatrix}\)

Compute the difference for each corresponding element:
- Subtract the first element in \(B\) (12) from the first in \(A\) (7), resulting in -5.
- Continue this process for all elements.

The result of this operation is a new matrix:\[A - B = \begin{bmatrix} -5 & -7 \ -3 & 13 \ 5 & -5 \end{bmatrix}\] which depicts the difference between the supplied data sets.

Scalar Multiplication

Scalar multiplication involves multiplying every element of a matrix by a scalar (a single number). This operation is straightforward but powerful, as it scales the entire matrix by the scalar value, altering the magnitude of its data.

For instance, given

• \(A = \begin{bmatrix} 7 & -4 \ -5 & 9 \ -1 & 2 \end{bmatrix}\)

if we multiply by a scalar of 2, each element in \(A\) is multiplied by 2. This results in the matrix:\[2A = \begin{bmatrix} 14 & -8 \ -10 & 18 \ -2 & 4 \end{bmatrix}\]Similarly, applying scalar multiplication to \(B\):

• \(B = \begin{bmatrix} 12 & 3 \ -2 & -4 \ -6 & 7 \end{bmatrix}\)

Multiplying by a scalar of 3 results in:\[3B = \begin{bmatrix} 36 & 9 \ -6 & -12 \ -18 & 21 \end{bmatrix}\] This operation efficiently magnifies the entire matrix by whatever number you've chosen as the scalar, whether that number is negative or positive.

Linear Algebra Expression Evaluation

Evaluating complex linear algebra expressions often involves combining several matrix operations, such as addition, subtraction, and scalar multiplication. This step enables us to derive meaningful results from intricate expressions.

For example, the expression \(2A + 3B\) involves both scalar multiplication and matrix addition:

• First multiply each element in \(A\) by 2 and \(B\) by 3 as described in scalar multiplication.
• Then, add the resulting matrices: \(2A = \begin{bmatrix} 14 & -8 \ -10 & 18 \ -2 & 4 \end{bmatrix}\) with \(3B = \begin{bmatrix} 36 & 9 \ -6 & -12 \ -18 & 21 \end{bmatrix}\).

This addition gives:\[2A + 3B = \begin{bmatrix} 50 & 1 \ -16 & 6 \ -20 & 25 \end{bmatrix}\]

Similarly, evaluate \(4A - 2B\) using scalar multiplication and subtraction:
- Calculate \(4A\) and \(2B\) then subtract:
\[4A - 2B = \begin{bmatrix} 4 & -22 \ -16 & 44 \ 8 & -6 \end{bmatrix}\]
Through these steps, linear algebra expressions are systematically reduced to simpler matrix forms, enabling deeper insights and understanding.