Problem 4 - Algebra and Trigonometry with Analytic Geometry

Question

Find, if possible, \(A+B, A-B, 2 A\), and \(-3 B\) $$ A=\left[\begin{array}{rrr} 0 & -2 & 7 \\ 5 & 4 & -3 \end{array}\right], \quad B=\left[\begin{array}{lll} 8 & 4 & 0 \\ 0 & 1 & 4 \end{array}\right] $$

Step-by-Step Solution

Answer

A+B=\begin{bmatrix}8 & 2 & 7 \\ 5 & 5 & 1 \end{bmatrix}, A-B=\begin{bmatrix}-8 & -6 & 7 \\ 5 & 3 & -7 \end{bmatrix}, 2A=\begin{bmatrix}0 & -4 & 14 \\ 10 & 8 & -6 \end{bmatrix}, -3B=\begin{bmatrix}-24 & -12 & 0 \\ 0 & -3 & -12 \end{bmatrix}.

Step 1: Understand Matrix Dimensions

First, verify the dimensions of matrices A and B. Matrix A is a 2x3 matrix and Matrix B is also a 2x3 matrix. Since matrix addition and subtraction require the same dimensions, both matrices fit the criteria.

Step 2: Add Matrices A and B

To find \(A + B\), add corresponding elements of A and B. This results in: \[A + B = \begin{bmatrix} 0+8 & -2+4 & 7+0 \ 5+0 & 4+1 & -3+4 \end{bmatrix} = \begin{bmatrix} 8 & 2 & 7 \ 5 & 5 & 1 \end{bmatrix}\]

Step 3: Subtract Matrix B from A

To find \(A - B\), subtract corresponding elements of B from A. This results in: \[A - B = \begin{bmatrix} 0-8 & -2-4 & 7-0 \ 5-0 & 4-1 & -3-4 \end{bmatrix} = \begin{bmatrix} -8 & -6 & 7 \ 5 & 3 & -7 \end{bmatrix}\]

Step 4: Multiply Matrix A by a Scalar

To find \(2A\), multiply each element of A by 2. This results in: \[2A = \begin{bmatrix} 2\cdot0 & 2\cdot(-2) & 2\cdot7 \ 2\cdot5 & 2\cdot4 & 2\cdot(-3) \end{bmatrix} = \begin{bmatrix} 0 & -4 & 14 \ 10 & 8 & -6 \end{bmatrix}\]

Step 5: Multiply Matrix B by a Scalar

To find \(-3B\), multiply each element of B by -3. This results in: \[-3B = \begin{bmatrix} -3\cdot8 & -3\cdot4 & -3\cdot0 \ -3\cdot0 & -3\cdot1 & -3\cdot4 \end{bmatrix} = \begin{bmatrix} -24 & -12 & 0 \ 0 & -3 & -12 \end{bmatrix}\]

Key Concepts

Matrix Addition

Matrix addition is one of the simplest operations you can perform with matrices.
The key requirement is that both matrices must be of the same dimensions.
In other words, they should have the same number of rows and columns.
For example, if both matrices are 2x3 matrices, like matrices A and B from our example, you can add them together.
When adding matrices, you simply add the corresponding elements from each matrix.
Think of it as lining up one matrix on top of the other and adding the boxes that match each other in position.
For example, in the matrix operation \(A + B\), you would perform the following calculations:

• For the first element in the first row: \(0 + 8 = 8\)
• For the second element in the first row: \(-2 + 4 = 2\)
• Continue this way for each element.

The result is a new matrix that has the same dimensions as the original matrices but with updated values.
It's straightforward and makes matrix algebra accessible to anyone.

Matrix Subtraction

Matrix subtraction is just as simple as matrix addition.
However, instead of adding corresponding elements, you subtract them.
Like addition, both matrices need to have the same dimensions.
This ensures that each element can be matched correctly for subtraction.Let's consider matrices A and B again.
To calculate \(A - B\), you subtract each element of matrix B from the corresponding element in matrix A:

• Start with the first element in the first row: \(0 - 8 = -8\)
• Then proceed to the second element of the first row: \(-2 - 4 = -6\)
• Continue until all elements have been subtracted.

The result is another matrix of the same dimensions as the original matrices, but with the newly computed values.
Like addition, matrix subtraction is a basic operation that builds foundational understanding in matrix algebra.

Scalar Multiplication

Scalar multiplication involves multiplying every element of a matrix by a number known as a scalar.
This operation can be done on any matrix, regardless of its dimensions.
It's one of the most straightforward operations involving matrices because it's essentially simple arithmetic.Take matrix A and a scalar value of 2, per our example.
To find \(2A\), you multiply each term in matrix A by 2:

• The first element in the first row becomes: \(2 \cdot 0 = 0\)
• The second element in the first row becomes: \(2 \cdot (-2) = -4\)
• Continue this way for all elements in the matrix.

The result is a new matrix where each element has been scaled by that factor.
This operation can be useful for transforming matrices and performing more complex operations later on.

Matrix Dimensions

Understanding matrix dimensions is crucial when learning matrix operations.
A matrix is defined by its number of rows and columns.
The dimensions are expressed as \(m \times n\), where \(m\) is the number of rows and \(n\) is the number of columns.For instance, in our example:

• Matrix A has dimensions of 2x3, meaning it has 2 rows and 3 columns.
• Matrix B also has dimensions of 2x3.

These dimensions are essential because they determine the feasibility of matrix operations like addition and subtraction.
Only matrices with the same dimensions can be added or subtracted.
Always double-check the dimensions before performing any operations to avoid errors.
This step helps ensure all elements align correctly, facilitating accurate results in your calculations.