Problem 18 - Algebra 2

Question

Perform the indicated matrix operations. If the matrix does not exist, write impossible. $$ \left[\begin{array}{rrr}{2} & {5} & {3} \\ {-7} & {-1} & {11} \\ {4} & {-4} & {0}\end{array}\right]+\left[\begin{array}{rrr}{-9} & {2} & {-5} \\ {1} & {6} & {-3} \\ {-9} & {-12} & {8}\end{array}\right] $$

Step-by-Step Solution

Answer

The resulting matrix is \[ \begin{bmatrix} -7 & 7 & -2 \\ -6 & 5 & 8 \\ -5 & -16 & 8 \end{bmatrix} \]

Step 1: Review matrix dimensions

First, check the dimensions of both matrices to ensure they can be added together. Both matrices are 3x3, which makes them compatible for addition.

Step 2: Add corresponding elements

Add each element in the first matrix to the corresponding element in the second matrix. This involves element-wise addition.

Step 3: Addition calculations

Perform the calculations for each position: Top left: \(2 + (-9) = -7\)Top middle: \(5 + 2 = 7\)Top right: \(3 + (-5) = -2\)Middle left: \(-7 + 1 = -6\)Middle middle: \(-1 + 6 = 5\)Middle right: \(11 + (-3) = 8\)Bottom left: \(4 + (-9) = -5\)Bottom middle: \(-4 + (-12) = -16\)Bottom right: \(0 + 8 = 8\)

Step 4: Write the resulting matrix

Combine all calculated elements into their respective positions to form the resulting matrix:\[\begin{bmatrix}-7 & 7 & -2 \-6 & 5 & 8 \-5 & -16 & 8\end{bmatrix}\]

Key Concepts

Matrix Operations

Matrix operations are a fundamental part of linear algebra, allowing us to manipulate matrices to solve various mathematical problems. Some common matrix operations include addition, subtraction, multiplication, and finding inverses. Matrix addition is one of the simplest operations and involves adding corresponding elements from two matrices. Before performing any operation, it's important to ensure that the matrices are compatible, which typically means having the same dimensions. When adding two matrices, each element in one matrix is added to the corresponding element in the second matrix to produce a new matrix with the same dimensions. Matrix operations facilitate numerous applications in fields such as computer graphics, physics, and statistics. As you continue learning, you'll encounter more complex matrix operations, but understanding the basics like matrix addition is essential for building a strong foundation in mathematics.

3x3 Matrices

3x3 matrices are one of the most common types of matrices you'll encounter in mathematical problems. They consist of three rows and three columns. They are a special kind of square matrix where the number of rows equals the number of columns, which frequently simplifies calculations and allows for specific properties not found in non-square matrices. In a 3x3 matrix, every element has a designated position, identified by its row and column. For example, the element in the first row and first column is distinct from those in any other position. When working with 3x3 matrices, it’s crucial to maintain the order of elements during operations to ensure accuracy. 3x3 matrices are used extensively in various calculations, such as solving systems of equations, computer graphics transformations, and representing vectors and their transformations. Understanding the structure of 3x3 matrices is essential for performing matrix operations like addition smoothly.

Element-wise Addition

Element-wise addition is the process of adding each element in one matrix to the corresponding element in another matrix. It is perhaps the most straightforward yet crucial operation in matrix algebra.To perform element-wise addition, both matrices must have the exact same dimensions, which means each one must have an equal number of rows and columns. If matrices don't match in size, addition is impossible. Each pair of corresponding elements from the two matrices is summed to form the elements of a new, resulting matrix.For instance, with 3x3 matrices, the element in the top-left corner of the first matrix adds to the top-left element of the second matrix, and so forth across the entire matrix:

• Top left: calculate \(2 + (-9) = -7\)
• Top right: calculate \(3 + (-5) = -2\)
• Middle middle: calculate \(-1 + 6 = 5\)
• Continue this pattern for each element.

Element-wise addition provides a way to combine matrices and is a stepping stone to understanding more complex operations like matrix multiplication. It’s a foundational skill in matrix algebra that enables you to handle more intricate matrix manipulations in mathematical applications.