Problem 7 - Precalculus Mathematics for Calculus

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 resulting matrix is \(\begin{array}{rr} 1 & 3 \\ 1 & 5 \end{array}\).

Step 1: Verify Matrix Dimensions

To add two matrices, they must have the same dimensions. Each matrix given in the problem is a 2x2 matrix, which means they both have 2 rows and 2 columns. Therefore, these matrices can be added together.

Step 2: Perform Element-wise Addition

Add the corresponding elements of the matrices together. Take the element in the first row, first column of each matrix and add them together, continue this for each element:\[\begin{array}{rr}2 & 6 \-5 & 3\end{array}+\begin{array}{rr}-1 & -3 \6 & 2\end{array}=\begin{array}{rr}2 + (-1) & 6 + (-3) \-5 + 6 & 3 + 2\end{array}\]

Step 3: Simplify Each Element

Calculate each element in the resulting 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: \(3 + 2 = 5\)Thus, the resulting matrix is:\[\begin{array}{rr}1 & 3 \1 & 5\end{array}\]

Key Concepts

Matrix Dimensions

Understanding matrix dimensions is crucial for working with matrices in mathematical operations. A matrix's dimensions refer to its size, described by the number of rows and columns it has. In mathematical notation, a matrix with two rows and two columns is called a 2x2 matrix.
For any matrix operation to be valid, particularly addition, the matrices involved must have the same dimensions. This ensures that each element in one matrix has a corresponding element in the other to pair with during the operation.
For instance, when considering two matrices:

• The first matrix is \(\left[\begin{array}{rr}2 & 6 \ -5 & 3\end{array}\right] \).
• The second matrix is \(\left[\begin{array}{rr}-1 & -3 \ 6 & 2\end{array}\right] \).

Both matrices share the same dimensions, being 2x2 matrices. This permits their addition.

Element-wise Operation

Matrix addition is achieved through an element-wise operation. This process involves the addition of corresponding elements from each matrix. Essentially, you add each element from one matrix to the element in the same position in the other matrix.
Here's picturing the concept:

• The top-left element of the matrices are added together: \(2 + (-1) = 1\).
• The top-right elements are combined: \(6 + (-3) = 3\).
• The bottom-left elements become: \(-5 + 6 = 1\).
• The bottom-right elements add up to: \(3 + 2 = 5\).

By performing this element-wise operation, you achieve a new matrix where each element is the sum of the corresponding elements from the original matrices.

2x2 Matrix

A 2x2 matrix is one of the simplest forms of a matrix, making it easy to manipulate and calculate in matrix operations such as addition, subtraction, and multiplication. Each 2x2 matrix consists of two rows and two columns, creating a total of four elements.
The structure of a 2x2 matrix is typically written as follows: \(\left[\begin{array}{cc}a & b \ c & d\end{array}\right] \). Here, \(a, b, c,\) and \(d\) are the individual elements of the matrix.
When dealing with matrix addition, 2x2 matrices provide a clear and straightforward example for understanding the concept because each element pairs directly with another corresponding element. This simplicity makes 2x2 matrices perfect for illustrating basic matrix operations.

Matrix Operations

Matrix operations involve various procedures that manipulate matrices to obtain different outcomes, such as addition, subtraction, and multiplication, based on the mathematical rules governing matrices.
Among these operations, matrix addition is one of the most fundamental. This operation can only be performed when the matrices involved have the same dimensions. Once matched, you apply element-wise addition.
The completed matrix operation in our example resulted in:

• Matrix before: \(\left[\begin{array}{rr}2 & 6 \ -5 & 3\end{array}\right] + \left[\begin{array}{rr}-1 & -3 \ 6 & 2\end{array}\right]\)
• Resulting matrix: \(\left[\begin{array}{rr}1 & 3 \ 1 & 5\end{array}\right]\)

Understanding and practicing these operations on simple matrices is fundamental for building up to more complex calculations in linear algebra.