Problem 14 - College Algebra

Question

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

Step-by-Step Solution

Answer

a. \(A+B = (10,0,0)\). b. \(A-B = (2,4,-6)\). c. \(-4A = (-24, -8, 12)\). d. \(3A+2B = (26,2,-3)\).

Step 1: Matrix Addition

Performing \( A + B \) addition, which is done element-wise. Thus, \((6 + 4, 2 + (-2), -3+3) = (10,0,0)\).

Step 2: Matrix Subtraction

Performing \( A - B \) subtraction, which is done element-wise. Thus, \((6 - 4, 2 - (-2), -3 - 3) = (2,4,-6)\).

Step 3: Scalar Matrix Multiplication

Performing \(-4 \times A\), which means every element of matrix A should be multiplied by -4. Thus, \(-4 \times (6,2,-3) = (-24, -8, 12)\).

Step 4: Scalar Matrix Multiplication and Addition

Multiplying matrices A and B by scalars and then adding them: \( 3 \times A + 2 \times B \). So, \(3 \times (6,2,-3) = (18, 6, -9)\) and \(2 \times (4,-2,3) = (8, -4, 6)\). Adding them together gives \((18 + 8, 6 + (-4), -9 + 6) = (26,2,-3)\).

Key Concepts

Matrix Addition

Matrix addition is a fundamental operation in linear algebra. To perform matrix addition, each element of one matrix is added to the corresponding element of another matrix. However, this operation is only possible if the matrices have the same dimensions. For example, in our exercise, the matrices A and B are both 1x3 matrices, which means they have one row and three columns.

Let's consider the given matrices:

• Matrix A is \[A=\begin{bmatrix}6 & 2 & -3\end{bmatrix}\].
• Matrix B is \[B=\begin{bmatrix}4 & -2 & 3\end{bmatrix}\].

To add these matrices:

• Add the first elements: 6 (from A) + 4 (from B) = 10.
• Add the second elements: 2 (from A) - 2 (from B, since it's negative) = 0.
• Add the third elements: -3 (from A) + 3 (from B) = 0.

Therefore, the result of A + B is a new 1x3 matrix: \[\begin{bmatrix}10 & 0 & 0\end{bmatrix}\].Understanding matrix addition is crucial as it forms the basis for more complex operations such as matrix multiplication and finding the solutions to linear equations.

Matrix Subtraction

Matrix subtraction is similar to matrix addition, but instead of adding, we subtract the corresponding elements of one matrix from another. As with addition, the matrices must be of the same dimensions to subtract them from each other. The exercise provides a clear example of how this operation works with 1x3 matrices.

Given the matrices A and B as before:

• Subtract the first elements: 6 (from A) - 4 (from B) = 2.
• Subtract the second elements: 2 (from A) - (-2) (from B, remember subtracting a negative is the same as adding) = 4.
• Subtract the third elements: -3 (from A) - 3 (from B) = -6.

This results in a new matrix: \[\begin{bmatrix}2 & 4 & -6\end{bmatrix}\].Notice that the order of subtraction is important. In contrast to addition, which is commutative, subtraction is not. So, A - B is not the same as B - A. This is an important concept to remember because it can impact the solution to a problem, especially when dealing with systems of equations or transformations in geometry.

Scalar Matrix Multiplication

Scalar matrix multiplication involves multiplying every element of a matrix by a single number, known as a scalar. This operation is used to scale a matrix up or down, and it does not change the dimension of the matrix. In our exercise, we encounter this operation twice, first when multiplying matrix A by -4, and then when performing a combination of scalar multiplication and addition.

Example of Scalar Multiplication

• Multiply each element of matrix A by -4: \[(-4 \times 6, -4 \times 2, -4 \times -3) = (-24, -8, 12)\].

Combining Operations

To solve the part (d) of the exercise, we first perform scalar multiplication on each matrix and then add the results:

• Multiply each element of matrix A by 3: \[(3 \times 6, 3 \times 2, 3 \times -3) = (18, 6, -9)\].
• Multiply each element of matrix B by 2: \[(2 \times 4, 2 \times -2, 2 \times 3) = (8, -4, 6)\].
• Add the results of the scalar multiplications: \[(18 + 8, 6 - 4, -9 + 6) = (26, 2, -3)\].

This demonstrates how matrices can be manipulated and combined using scalar multiplication, laying the foundation for complex matrix operations like matrix multiplication and determinants.