Problem 7 - Algebra and Trigonometry

Question

Perform the matrix operation, or if it is impossible, explain why. $$ \left[\begin{array}{rr}{2} & {6} \\ {-5} & {3}\end{array}\right]+\left[\begin{array}{rr}{-1} & {-3} \\ {6} & {2}\end{array}\right] $$

Step-by-Step Solution

Answer

The matrices can be added to get: \(\begin{bmatrix} 1 & 3 \\ 1 & 5 \end{bmatrix}\).

Step 1: Verify Matrix Dimensions

Both matrices are 2x2 matrices. Matrix addition is valid when both matrices have the same dimensions. Since both are 2x2, we can proceed with the addition.

Step 2: Add Corresponding Elements

To add the matrices, add each element from the first matrix to the corresponding element in the second matrix. - First row, first column: \(2 + (-1) = 1\)- First row, second column: \(6 + (-3) = 3\)- Second row, first column: \(-5 + 6 = 1\)- Second row, second column: \(3 + 2 = 5\)

Step 3: Write the Resulting Matrix

Place each result from Step 2 into the corresponding position of a new 2x2 matrix: \[\begin{bmatrix}1 & 3 \1 & 5\end{bmatrix}\]

Key Concepts

Matrix Dimensions

Matrix dimensions are essential when working with matrices in mathematical operations. A matrix is usually defined by its number of rows and columns, noted as "rows x columns." These dimensions determine how matrices can interact with each other, especially in operations like addition and multiplication.

For matrix addition, it is crucial that both matrices have the same dimensions. This means that they have the same number of rows and columns. If they don't match, the operation cannot be performed. It's like trying to add two lists that have different lengths—it simply doesn't work because there's no alignment of elements to add.

For example, if you have a matrix with dimensions 2x2, it must be added to another matrix with dimensions 2x2 to carry out matrix addition. This ensures each element across both matrices has a corresponding position to be added together.

2x2 Matrix

A 2x2 matrix is a small square matrix containing two rows and two columns. Each element in the matrix is organized into its distinct position based on its row and column location.

For example, consider a 2x2 matrix as follows:
\[\begin{bmatrix}a & b \c & d\end{bmatrix}\]
Here, elements \(a\), \(b\), \(c\), and \(d\) are arranged such that \(a\) and \(b\) form the first row, while \(c\) and \(d\) make up the second row. Similarly, \(a\) and \(c\) are in the first column, and \(b\) and \(d\) are in the second column.

2x2 matrices are very straightforward to work with in matrix operations, often serving as a good starting point for learning matrix arithmetic. Despite their simplicity, they frequently appear in various mathematical concepts and applications.

Matrix Operations

Matrix operations encompass various tasks such as addition, subtraction, multiplication, and more. Each of these operations involves specific rules and requirements.

**Matrix Addition:**
Matrix addition involves taking two matrices of the same dimensions and adding their corresponding elements. The process is quite straightforward but requires attention to detail to ensure accuracy.

• Check the dimensions: Only add matrices if dimensions match.
• Add corresponding elements: \(a_{ij} + b_{ij}\) gives you the element in the new matrix.
• Write the resulting matrix: Arrange summed elements in the same positions as in the original matrices.

In our original exercise, we saw a 2x2 matrix addition where each element from the first matrix was added to its corresponding element in the second matrix, yielding a new 2x2 matrix with each summed element placed in its respective position. Understanding these operations provides crucial skills for more complex matrix manipulations.