Problem 14 - Algebra and Trigonometry

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

The resulting matrices are: \(A + B = [10, 0, 0]\), \(A - B = [2, 4, -6]\), \(-4A = [-24, -8, 12]\), and \(3A + 2B = [26, 2, -3]\)

Step 1: Add Matrices A and B

To find \(A+B\), simply add corresponding elements from each matrix. This gives us a new matrix: \(A + B = [6+4, 2+(-2), -3+3] = [10, 0, 0]\)

Step 2: Subtract Matrix B from A

To find \(A-B\), subtract corresponding elements in B from corresponding elements in A. This gives us a new matrix: \(A - B = [6-4, 2-(-2), -3-3] = [2, 4, -6]\)

Step 3: Perform Scalar Multiplication on Matrix A

To find \(-4A\), simply multiply every element in matrix A by -4. This gives us a new matrix: \(-4A = -4*[6, 2, -3] = [-24, -8, 12]\)

Step 4: Combine Scalar Multiplication and Addition

To find \(3A + 2B\), first multiply every element in matrix A by 3 and every element in matrix B by 2, then add corresponding elements from each of the results. This gives us a new matrix: \(3A + 2B = 3*[6, 2, -3] + 2*[4, -2, 3] = [18, 6, -9] + [8, -4, 6] = [26, 2, -3]\)

Key Concepts

Matrix Addition

Matrix addition is the process of adding two matrices together. This operation is only possible when the matrices have the same dimensions. To add two matrices, you simply add the elements that are in the same position within each matrix.

For example, consider matrices A and B:

• Matrix A is: \( \begin{bmatrix} 6 & 2 & -3 \end{bmatrix} \)
• Matrix B is: \( \begin{bmatrix} 4 & -2 & 3 \end{bmatrix} \)

To perform the addition \(A + B\), we add each pair of corresponding elements from matrices A and B. The calculations are simple:

• Add the first elements: \(6 + 4 = 10\)
• Add the second elements: \(2 + (-2) = 0\)
• Add the third elements: \(-3 + 3 = 0\)

So, the resulting matrix from \(A + B\) is \( \begin{bmatrix} 10 & 0 & 0 \end{bmatrix} \). Notice how the operation is straightforward and requires only basic arithmetic!

Matrix Subtraction

Matrix subtraction is a bit like matrix addition, except that, as the name suggests, you subtract corresponding elements instead of adding them. Just like with addition, both matrices must have the same dimensions to perform subtraction.

In the given example, subtract matrix B from matrix A as follows:

• Matrix A: \( \begin{bmatrix} 6 & 2 & -3 \end{bmatrix} \)
• Matrix B: \( \begin{bmatrix} 4 & -2 & 3 \end{bmatrix} \)

For \(A - B\), the operation consists of:

• Subtract the first elements: \(6 - 4 = 2\)
• Subtract the second elements: \(2 - (-2) = 4\)
• Subtract the third elements: \(-3 - 3 = -6\)

Thus, the matrix resulting from \(A - B\) is \( \begin{bmatrix} 2 & 4 & -6 \end{bmatrix} \). Subtraction is similarly straightforward, just requiring a cautious approach to manage negative numbers and avoid simple arithmetic errors.

Scalar Multiplication of Matrices

Scalar multiplication is the operation of multiplying each element of a matrix by a constant number, called a scalar. This operation doesn't require the matrices to be of any specific size as each element is independently multiplied by the scalar.

Let's look at the example with matrix A and a scalar \(-4\):

• Matrix A: \( \begin{bmatrix} 6 & 2 & -3 \end{bmatrix} \)
• Scalar: \(-4\)

To perform \(-4A\), multiply each element of matrix A by \(-4\):

• The first element: \(-4 \times 6 = -24\)
• The second element: \(-4 \times 2 = -8\)
• The third element: \(-4 \times (-3) = 12\)

The new matrix resulting from this operation is \( \begin{bmatrix} -24 & -8 & 12 \end{bmatrix} \). Scalar multiplication is a straightforward way to change all the elements in a matrix by a constant factor and is commonly used in more complex matrix calculations.