Problem 21 - Algebra and Trigonometry Real Mathematics, Real People

Question

Evaluating an Expression Evaluate the expression. $$\left[\begin{array}{rr} -5 & 0 \\ 3 & -6 \end{array}\right]+\left[\begin{array}{rr} 7 & 1 \\ -2 & -1 \end{array}\right]+\left[\begin{array}{rr} -10 & -8 \\ 14 & 6 \end{array}\right]$$

Step-by-Step Solution

Answer

The result of the matrix addition is: \[\left[\begin{array}{rr} -8 & -7 \ 15 & -1 \end{array}\right]\]

Step 1: Add the Elements in the First Row and First Column

Add the first elements from each matrix: \(-5 + 7 - 10\). This gives the first number of the resultant matrix.

Step 2: Add the Elements in the First Row and Second Column

Next add the second elements from the first row of each matrix: \(0 + 1 - 8\). This gives the second number in the first row of the resultant matrix.

Step 3: Add the Elements in the Second Row and First Column

The next step is to add the first elements from the second row of each matrix: \(3 - 2 + 14\). This gives the first number in the second row of the resultant matrix.

Step 4: Add the Elements in the Second Row and Second Column

Finally add the second elements from the second row of each matrix: \(-6 -1 + 6\). This gives the second number in the second row of the resultant matrix

Key Concepts

Element-wise Addition

In matrix addition, one of the fundamental concepts to grasp is **element-wise addition**. This idea simply means that you add corresponding elements from each matrix to obtain a new set of values in the resulting matrix. Each matrix must be of the same dimensions so that every element in one matrix has a matching element in the other matrices. For our problem, we're looking at matrices that each have two rows and two columns.

To perform element-wise addition:

• Add the numbers found in the same position of each matrix together.
• Proceed row by row and column by column to maintain the order.

For example, considering the matrix positions, the element in the first row and first column from each matrix needs to be added together. With our exercise, this means \[-5+7-10\] for the first position. Follow this pattern for each subsequent position.

Resultant Matrix

The outcome of element-wise addition is what we call the **resultant matrix**. This matrix contains values that result from the addition of each corresponding element from the matrices being combined. In essence, the resultant matrix holds the solution to the matrix addition operation.

When we calculated each position:

• The first row, first column is \[-5+7-10=-8\].
• The first row, second column becomes \[0+1-8=-7\].
• Moving to the second row, the first column results in \[3-2+14=15\].
• And finally, the second row, second column is \[-6-1+6=-1\].

This process gives us our resultant matrix:\[\begin{bmatrix}-8 & -7 \15 & -1\end{bmatrix}\]This is the matrix that represents the combined values derived from each individual element-wise addition.

Matrix Operations

Understanding the broader topic of **matrix operations** is important when tackling problems like matrix addition. Matrix operations are fundamental procedures applied to matrices, allowing us to manipulate and derive useful information from them. Alongside addition, common matrix operations include subtraction, multiplication, and transformation.

For matrix addition, remember these key points:

• Matrices must be of the same dimensions for accurate addition.
• Each operation needs to maintain the integrity and position of elements.
• The resultant matrix inherits these dimensions.

These operations not only apply in theoretical math but also have practical applications in computer graphics, scientific computations, and more. Understanding these basic operations sets the foundation for more complex matrix manipulations and applications.