Problem 16 - College Algebra

Question

In Exercises \(9-16,\) find: a. \(A+B\) b. \(A-B\) c. \(-4 A\) d. \(3 A+2 B\) $$ A=\left[\begin{array}{rrr} 6 & -3 & 5 \\ 6 & 0 & -2 \\ -4 & 2 & -1 \end{array}\right], \quad B=\left[\begin{array}{rrr} -3 & 5 & 1 \\ -1 & 2 & -6 \\ 2 & 0 & 4 \end{array}\right] $$

Step-by-Step Solution

Answer

a. \(A + B = \left[\begin{array}{rrr} 3 & 2 & 6 \ 5 & 2 & -8 \ -2 & 2 & 3 \end{array}\right]\) \n b. \(A - B = \left[\begin{array}{rrr} 9 & -8 & 4 \ 7 & -2 & 4 \ -6 & 2 & -5 \end{array}\right]\) \n c. \(-4*A = \left[\begin{array}{rrr} -24 & 12 & -20 \ -24 & 0 & 8 \ 16 & -8 & 4 \end{array}\right]\) \n d. \(3 * A + 2 * B = \left[\begin{array}{rrr} 6 & -1 & 17 \ 16 & 6 & -22 \ -8 & 6 & -1 \end{array}\right]\)

Step 1: Find A + B

Add matrices A and B together. This is done by adding the corresponding elements from each matrix. For example, to find the element in the first row and first column of A + B, add the element in the first row and first column of A to the element in the first row and first column of B. Do this for each element in both matrices.

Step 2: Find A - B

Subtract matrix B from matrix A. This is done by subtracting the corresponding elements of matrix B from matrix A. Each element in the A - B matrix is the result of subtracting the corresponding element of matrix B from matrix A.

Step 3: Find -4 * A

Multiply each element in matrix A by -4. The resulting matrix is the scalar multiplication of matrix A by -4.

Step 4: Find 3 * A + 2 * B

Multiply each element in matrix A by 3 and each element in matrix B by 2. Add these two resulting matrices together. This would give us the matrix 3*A + 2*B.

Key Concepts

Matrix Addition

Matrix addition is a basic operation where two matrices of the same dimensions are added together by combining corresponding elements.When we add two matrices, say matrix \( A \) and matrix \( B \), we essentially take each element from the same position in the matrices and add them together.For example, in the exercise, the matrices \( A \) and \( B \) are given as:
\[ A = \begin{bmatrix} 6 & -3 & 5 \ 6 & 0 & -2 \ -4 & 2 & -1 \end{bmatrix}, \quad B = \begin{bmatrix} -3 & 5 & 1 \ -1 & 2 & -6 \ 2 & 0 & 4 \end{bmatrix} \]

• To find the first element of \( A + B \), add 6 (from \( A \)) and -3 (from \( B \)), resulting in 3.
• This process continues for all elements across the matrices, resulting in a new matrix.
• The complete addition would be calculated for every corresponding pair of elements from \( A \) and \( B \).

Thus, matrix addition involves straightforward arithmetic addition on an element-wise basis across matrices.

Matrix Subtraction

Matrix subtraction involves element-wise subtraction between correspondingly placed elements from two matrices of identical dimensions.Here, when you subtract matrix \( B \) from matrix \( A \), each element in matrix \( B \) is subtracted from the corresponding element in matrix \( A \).Continuing with the matrices from the exercises:

• Starting with the first element, subtract -3 (from \( B \)) from 6 (from \( A \)), resulting in 9.
• This subtraction is repeated for every element across both matrices.
• The resulting matrix will have elements representing the difference between corresponding elements of \( A \) and \( B \).

Matrix subtraction, like addition, requires each pair of corresponding elements to be operated upon separately, making the dimensions of the two matrices a crucial factor.

Scalar Multiplication

Scalar multiplication involves multiplying every element of a matrix by a scalar or a constant value.This operation transforms the entire matrix by a uniform scaling factor.In the exercise, we multiply matrix \( A \):

• Each element in \( A \) is multiplied by \(-4\).
• For example, the element 6 in \( A \) becomes -24 after the multiplication.
• This is applied to every element, changing the matrix consistently without altering its structure.

Scalar multiplication effectively scales the matrix's values, influencing its size and balance while preserving the original structure.

Linear Combination of Matrices

A linear combination of matrices involves creating a new matrix by multiplying existing matrices by scalars and then adding the results.The exercise incorporates such a concept by using matrices \( A \) and \( B \):

• Matrix \( A \) is multiplied by 3, and matrix \( B \) is multiplied by 2.
• Every element in matrix \( A \) is scaled and calculated separately from those in \( B \).
• The resulting matrices from these operations are then added together to form the final matrix.

This operation reflects a combination where both scaling and addition influence the final outcome, providing a new matrix derived from the originals but significantly transformed by the scalar coefficients.