Problem 14 - College Algebra

Question

Find the following matrices: a. \(A+B\) b. \(A-B\) c. \(-4 A\) d. \(3 A+2 B\) $$A=\left[\begin{array}{lll}6 & 2 & -3\end{array}\right], B=\left[\begin{array}{lll}4 & -2 & 3\end{array}\right]$$

Step-by-Step Solution

Answer

\(A+B=\left[\begin{array}{ccc}10 & 0 & 0\end{array}\right], A-B=\left[\begin{array}{ccc}2 & 4 & -6\end{array}\right], -4A=\left[\begin{array}{ccc}-24 & -8 & 12\end{array}\right], 3A+2B=\left[\begin{array}{ccc}26 & 2 & -3\end{array}\right]\)

Step 1: Addition of matrices A and B

Add each corresponding entry in matrices A and B. This gives the new matrix: \(A+B=\left[\begin{array}{ccc}6+4 & 2-2 & -3+3\end{array}\right]=\left[\begin{array}{ccc}10 & 0 & 0\end{array}\right]\)

Step 2: Subtraction of matrix B from A

Subtract each corresponding entry in matrix B from matrix A. This gives the new matrix: \(A-B=\left[\begin{array}{ccc}6-4 & 2+2 & -3-3\end{array}\right]=\left[\begin{array}{ccc}2 & 4 & -6\end{array}\right]\)

Step 3: Multiplication of matrix A by -4

Multiply each entry in matrix A by -4. This gives the new matrix: \(-4A=\left[\begin{array}{ccc}-4*6 & -4*2 & -4*-3\end{array}\right]=\left[\begin{array}{ccc}-24 & -8 & 12\end{array}\right]\)

Step 4: Linear combination of matrices

Multiply matrix A by 3 and matrix B by 2, respectively. Then, add the corresponding entries from the newly obtained matrices. This gives: \(3A+2B=\left[\begin{array}{ccc}3*6+2*4 & 3*2+2*(-2) & 3*-3+2*3\end{array}\right]=\left[\begin{array}{ccc}26 & 2 & -3\end{array}\right]\)

Key Concepts

Matrix Addition

Matrix addition is a straightforward operation. Think of it like simple arithmetic addition but done step-by-step for each element in two matrices. For this operation to work, both matrices must have the same dimensions. In other words, the number of rows and columns should match in both matrices.

When you're adding matrices, take the element from the first row and the first column of the first matrix and add it to the element in the same position in the second matrix. Repeat this process for all elements.

• Example: Given matrix \( A = \begin{bmatrix} 6 & 2 & -3 \end{bmatrix} \) and matrix \( B = \begin{bmatrix} 4 & -2 & 3 \end{bmatrix} \), their sum is found by adding their corresponding elements: \( A+B = \begin{bmatrix} 6+4 & 2-2 & -3+3 \end{bmatrix} = \begin{bmatrix} 10 & 0 & 0 \end{bmatrix} \).

Matrix addition is thus just about adding each pair of elements individually.

Matrix Subtraction

Matrix subtraction is very much like addition, but instead, you subtract elements from one another. Again, the number of rows and columns should be the same in both matrices.

To perform matrix subtraction, take each element in the first matrix and subtract its corresponding element in the second matrix. It's just like doing subtraction in school but element by element.

• Example: Starting with matrix \( A = \begin{bmatrix} 6 & 2 & -3 \end{bmatrix} \) and matrix \( B = \begin{bmatrix} 4 & -2 & 3 \end{bmatrix} \), their difference is computed by subtracting the elements: \( A-B = \begin{bmatrix} 6-4 & 2-(-2) & -3-3 \end{bmatrix} = \begin{bmatrix} 2 & 4 & -6 \end{bmatrix} \).

Subtracting matrices simply involves swapping out addition for subtraction at every step.

Scalar Multiplication

Scalar multiplication involves multiplying every element in a matrix by a single number, called the scalar. This operation is simpler because it only consists of a single type of operation—multiplication—across all elements.

Why is scalar multiplication helpful? It allows you to scale all the elements of a matrix, which can be very useful in various mathematical and practical scenarios.

• Example: Let's consider multiplying matrix \( A = \begin{bmatrix} 6 & 2 & -3 \end{bmatrix} \) by the scalar \(-4\). This involves multiplying each element of \( A \) by \(-4\), so \(-4A = \begin{bmatrix} -4 \times 6 & -4 \times 2 & -4 \times (-3) \end{bmatrix} = \begin{bmatrix} -24 & -8 & 12 \end{bmatrix} \).

Scalar multiplication retains the structure but changes the magnitude of each matrix element.

Linear Combination

In linear algebra, a linear combination involves combining matrices through both scalar multiplication and addition or subtraction. You'll often encounter this when solving systems of equations or transforming vectors.

To create a linear combination of two matrices, like \(3A + 2B\), you first apply scalar multiplication to each matrix, then perform matrix addition on the results.

• Example: If you have matrix \( A = \begin{bmatrix} 6 & 2 & -3 \end{bmatrix} \) and matrix \( B = \begin{bmatrix} 4 & -2 & 3 \end{bmatrix} \), the linear combination \(3A + 2B\) is calculated by multiplying \( A \) by 3, \( B \) by 2, and then adding the results. Calculate it as: \( 3A + 2B = \begin{bmatrix} 3 \times 6 + 2 \times 4 & 3 \times 2 + 2 \times (-2) & 3 \times (-3) + 2 \times 3 \end{bmatrix} = \begin{bmatrix} 26 & 2 & -3 \end{bmatrix} \).

Creating linear combinations is a powerful tool in manipulating and utilizing matrices.