Problem 52 - Algebra 2

Question

Add or subtract. $$ \left[\begin{array}{cc}{5} & {-3} \\ {4} & {11}\end{array}\right]+\left[\begin{array}{rr}{4} & {0} \\ {-9} & {1}\end{array}\right] $$

Step-by-Step Solution

Answer

The resulting matrix after addition is \( \left[\begin{array}{cc}{9} & {-3} \ {-5} & {12}\end{array}\right] \).

Step 1: Identify Like Terms

Identify corresponding elements from each matrix that will be added together.

Step 2: Perform Element-wise Addition

Add the corresponding elements from the two matrices.

Step 3: Write the Resulting Matrix

Combine the sums of corresponding elements into a new matrix, keeping the same dimensions.

Key Concepts

Algebra 2

In the realm of Algebra 2, working with matrices falls under a broader category of operations that involve handling complex equations and systems of equations. These operations include matrix addition, among other things, allowing students to solve linear algebra problems more efficiently.

Students learn to manipulate various mathematical objects much like numbers, except that matrices are grids of numbers. Matrix operations extend the basic principles of algebra and arithmetic to two-dimensional planes, preparing students for higher mathematics and problem-solving in fields like engineering, economics, and computer science.

Matrices

Understanding Matrices

Matrices are rectangular arrays of numbers or functions arranged in rows and columns. They serve as a concise way of representing and working with multiple linear equations. In academic courses like Algebra 2, matrices are integral as they offer a pathway to manage data sets and perform calculations for linear transformations, systems of equations, and much more.

Each number in a matrix is known as an element. When adding matrices, it's crucial to understand that only matrices of the same dimensions can be added—that is, they must have the same number of rows and columns.

Elementary Operations

Matrix Addition and Subtraction

Elementary operations in linear algebra include simple tasks like matrix addition and subtraction. These tasks are important as they form the building blocks for more complex operations like multiplication and determining inverses. For matrix addition, you pairwise add the corresponding elements of the matrices. The element in the first row and first column of the first matrix is added to the element in the same position in the second matrix, and this operation is performed for each element.

The assignment to add matrices straightforwardly translates to adding each element with its counterpart, proceeding in an element-wise fashion. This parallels the way we add numbers, but it requires careful alignment of each corresponding element.

Linear Algebra

Importance of Linear Algebra in Matrix Operations

Linear algebra is the bedrock on which matrix theory is built. Understanding matrix addition is fundamental in mastering linear algebra as it deals with the linear combination of vectors, which are represented by matrices. A vector can be thought of as a matrix with only one column when it comes to the column vector form.

Adding matrices corresponds to adding vectors in a multidimensional space. When matrices are used in linear algebra, they represent complex systems that can model real-world problems. This operation is not merely arithmetic; it is the synchronization of data, the alignment of equations, and the simplification of computation.