Problem 8 - College Algebra and Calculus: An Applied Approach

Question

Find (a) \(A+B\), (b) \(A-B\), (c) \(3 A\), and (d) \(3 A-2 B\). $$ \begin{aligned} &A=\left[\begin{array}{rrrrr} 6 & 8 & -3 & 2 & 1 \\ -4 & 2 & 1 & 5 & -2 \end{array}\right], \\ &B=\left[\begin{array}{llrrr} 6 & 0 & 4 & -1 & 3 \\ 4 & 5 & -2 & 1 & 2 \end{array}\right] \end{aligned} $$

Step-by-Step Solution

Answer

The solutions are: (a) Matrix A plus B is \([12, 8, 1, 1, 4], [0, 7, -1, 6, 0]\). (b) Matrix A minus B is \([0, 8, -7, 3, -2], [-8, -3, 3, 4, -4]\). (c) Three times Matrix A is \([18, 24, -9, 6, 3], [-12, 6, 3, 15, -6]\). (d) Matrix \(3A - 2B\) is \([6, 24, -17, 8, -3], [-20, -4, 5, 12, -10]\).

Step 1: Add A and B

Find the sum of the matrices A and B. To do this, add together corresponding elements in the matrices: \( (6+6, 8+0, -3+4, 2-1, 1+3) \) and \(-4+4, 2+5, 1-2, 5+1, -2+2) \)

Step 2: Subtract B from A

Find the difference of the matrices A and B. To do this, subtract the elements of B from the corresponding elements of A: \( (6-6, 8-0, -3-4, 2+1, 1-3) \) et \(-4-4, 2-5, 1+2, 5-1, -2-2) \)

Step 3: Multiply A by 3

Now, find the result of multiplying matrix A by 3. To do this, multiply every element of A by 3: \((3*6, 3*8, 3*(-3), 3*2, 3*1) \) and \((3*(-4), 3*2, 3*1, 3*5, 3*(-2))\)

Step 4: Find 3A - 2B

Finally, to find \(3A-2B\), subtract twice matrix B from three times matrix A. Both operations are similar to previous steps.

Key Concepts

Matrix Addition

Matrix addition is a simple operation that involves adding corresponding elements of two matrices. This operation is only possible when the matrices have the same dimensions. To add two matrices, such as Matrix A and Matrix B, simply add the respective elements from each matrix. For example, if matrix A is:

• Row 1: [6, 8, -3, 2, 1]
• Row 2: [-4, 2, 1, 5, -2]

and matrix B is:

• Row 1: [6, 0, 4, -1, 3]
• Row 2: [4, 5, -2, 1, 2]

The resulting matrix, after adding A and B, will have elements calculated like this: - First element of both matrices: 6 + 6 = 12, - Second element: 8 + 0 = 8, - Continue this process for each pair of elements.

Matrix Subtraction

Matrix subtraction works much like addition, but instead of adding corresponding elements, you subtract them. Matrix subtraction also requires both matrices to have the same dimensions.
For matrix A and B given above, to perform A - B, subtract each corresponding element of matrix B from matrix A. For example:

• First element: 6 - 6 = 0,
• Second element: 8 - 0 = 8,
• Continue this process for each pair of elements.

This operation results in another matrix of the same size, but with each element adjusted by the subtraction of each pair.

Scalar Multiplication

Scalar multiplication involves multiplying every element of a matrix by a constant, called a scalar. This operation does not depend on the dimensions of the matrix and can be applied to any matrix.
For example, multiplying matrix A by 3 means that each element of A will be tripled. If A is:

• Row 1: [6, 8, -3, 2, 1]
• Row 2: [-4, 2, 1, 5, -2]

Then multiplying A by 3 would give:

• Row 1: [(3*6), (3*8), (3*-3), (3*2), (3*1)]
• Row 2: [(3*-4), (3*2), (3*1), (3*5), (3*-2)]

Each element is the result of the product of the scalar (3 in this case) and the original element value.

Linear Combinations

A linear combination, in the context of matrices, involves performing operations such as scalar multiplication and addition or subtraction on matrices to produce a new matrix. It combines various matrices by using scalars to scale them first.
From the original exercise, to find a linear combination like \(3A - 2B\), we must perform two steps:

• First, perform scalar multiplication on matrix A by 3.
• Then, multiply matrix B by 2.
• Finally, subtract the resulting scaled matrix 2B from the scaled matrix 3A.

This combination allows us to produce new solutions or adjustments to the matrices we are working with, reflecting the combined weighted effects of two or more matrices.