Problem 13 - Algebra and Trigonometry

Question

Find the following matrices: a. \(A+B\) b. \(A-B\) c. \(-4 A\) d. \(3 A+2 B\) $$ A=\left[\begin{array}{r} {2} \\ {-4} \\ {1} \end{array}\right], \quad B=\left[\begin{array}{r} {-5} \\ {3} \\ {-1} \end{array}\right] $$

Step-by-Step Solution

Answer

a. \(A + B = \left[\begin{array}{r} -3 \ -1 \ 0 \end{array}\right]\) \nb. \(A - B = \left[\begin{array}{r} 7 \ -7 \ 2 \end{array}\right]\) \nc. \(-4A = \left[\begin{array}{r} -8 \ 16 \ -4 \end{array}\right]\) \nd. \(3A + 2B = \left[\begin{array}{r} -4 \ -6 \ 1 \end{array}\right]\)

Step 1 - Calculate \(A + B\)

First, add corresponding elements of matrices A and B together: \(A + B = \left[\begin{array}{r} (2 + -5) \ (-4 + 3) \ (1 + -1) \end{array}\right] = \left[\begin{array}{r} -3 \ -1 \ 0 \end{array}\right]\).

Step 2: Calculate \(A - B\)

Subtract corresponding elements of matrix B from matrix A: \(A - B = \left[\begin{array}{r} (2 - -5) \ (-4 - 3) \ (1 - -1) \end{array}\right] = \left[\begin{array}{r} 7 \ -7 \ 2 \end{array}\right]\).

Step 3: Calculate \(-4*A\)

Multiply each element of matrix A by -4: \(-4*A = -4*\left[\begin{array}{r} 2 \ -4 \ 1 \end{array}\right] = \left[\begin{array}{r} -8 \ 16 \ -4 \end{array}\right]\).

Step 4: Calculate \(3A + 2B\)

First multiply each matrix by their corresponding scalar, then add them together: \(3A + 2B = 3*\left[\begin{array}{r} 2 \ -4 \ 1 \end{array}\right] + 2*\left[\begin{array}{r} -5 \ 3 \ -1 \end{array}\right] = \left[\begin{array}{r} 6 \ -12 \ 3 \end{array}\right] + \left[\begin{array}{r} -10 \ 6 \ -2 \end{array}\right] = \left[\begin{array}{r} -4 \ -6 \ 1 \end{array}\right]\)

Key Concepts

addition of matrices

Adding matrices is a fundamental operation in linear algebra, where you combine two matrices to form a new matrix. The process is simple if you remember that only matrices of the same size can be added together. This means they must have the same number of rows and columns.
To add two matrices, you sum their corresponding elements:

• Place the numbers from the same position in each matrix together.
• Add them to get the new element for the resulting matrix.

For example, consider matrices A and B:
\[ A = \begin{bmatrix} 2 \ -4 \ 1 \end{bmatrix}, \quad B = \begin{bmatrix} -5 \ 3 \ -1 \end{bmatrix} \]
To calculate \( A + B \) you perform the following steps:\[(2 + -5), (-4 + 3), (1 + -1) = \begin{bmatrix} -3 \ -1 \ 0 \end{bmatrix}\]
This result shows how each corresponding element in matrices A and B combine to form the final matrix.

subtraction of matrices

Subtracting matrices is quite similar to adding them. The rule remains that only matrices of the same dimensions can participate in subtraction. This operation helps you find the difference between matrices, element by element.
Here's how you subtract two matrices:

• Identify corresponding elements in each matrix.
• Subtract the element of the second matrix from the element of the first matrix.

With matrices A and B:\[ A = \begin{bmatrix} 2 \ -4 \ 1 \end{bmatrix}, \quad B = \begin{bmatrix} -5 \ 3 \ -1 \end{bmatrix} \]
You find \( A - B \) as follows:\[(2 - -5), (-4 - 3), (1 - -1) = \begin{bmatrix} 7 \ -7 \ 2 \end{bmatrix}\]
By ensuring you subtract each corresponding pair of elements appropriately, you solve the subtraction correctly.

scalar multiplication

Scalar multiplication involves multiplying each element of a matrix by a scalar, which is simply a real number. This operation is important when you need to scale matrices up or down.
For scalar multiplication:

• Multiply every element in the matrix by the given scalar.
• This transforms the values in the matrix accordingly but maintains its original shape.

Consider matrix A and the scalar -4:
\[ A = \begin{bmatrix} 2 \ -4 \ 1 \end{bmatrix} \]
When you multiply \(-4\) with matrix A:
\[-4 * A = \begin{bmatrix} -8 \ 16 \ -4 \end{bmatrix}\]
This operation shows how each element in matrix A has been multiplied by -4, changing their values yet keeping the matrix's structure intact.

vector matrices

Vector matrices are a special type of matrix with only one column, effectively making them a column vector. These matrices simplify operations and are frequently used in linear algebra.
Understanding vector matrices:

• Consist of a single column and multiple rows.
• Can represent vectors in coordinate systems.
• They are matrices with clear geometrical interpretations and applications.

Let's look at matrix A:
\[ A = \begin{bmatrix} 2 \ -4 \ 1 \end{bmatrix} \]
This is a vector matrix since it only has one column. In operations like addition and scalar multiplication, you treat it just like any other matrix, but the results offer insights into vector transformations and space analysis.