Problem 35 - Algebra 2

Question

Use matrices \(A=\left[\begin{array}{rrr}{5} & {7} & {3} \\ {-1} & {0} & {-4}\end{array}\right]\) and \(C=\left[\begin{array}{rrr}{-7} & {4} & {2} \\\ {1} & {-2} & {-3}\end{array}\right]\) for Exercises 35 and 36 What is the sum \(A+C ?\) A. The matrices cannot be added. \(\mathbf{B} \cdot\left[\begin{array}{rrr}{-2} & {11} & {5} \\ {0} & {-2} & {-7}\end{array}\right]\) \(C .\left[\begin{array}{ccc}{12} & {3} & {1} \\ {-2} & {2} & {-1}\end{array}\right]\) \(D \cdot\left[\begin{array}{ccc}{-35} & {28} & {6} \\ {-1} & {0} & {12}\end{array}\right]\)

Step-by-Step Solution

Answer

B. \(A+C = \left[\begin{array}{rrr}{-2} & {11} & {5} \ {0} & {-2} & {-7}\end{array}\right]\).

Step 1: Understand Matrix Addition

To add two matrices, they must have the same dimensions. Then, add the corresponding entries from the two matrices together to form a new matrix.

Step 2: Verify the Dimensions

Check if matrices A and C have the same dimensions. Both matrices A and C are of dimension 2x3, so they can be added together.

Step 3: Add Corresponding Entries

Add the corresponding entries from matrices A and C to find matrix A+C.\(A+C = \left[\begin{array}{rrr} 5 + (-7) & 7 + 4 & 3 + 2 \ -1 + 1 & 0 + (-2) & -4 + (-3) \end{array}\right]\).This results in\(A+C = \left[\begin{array}{rrr} -2 & 11 & 5 \ 0 & -2 & -7 \end{array}\right]\).

Key Concepts

Matrices

Imagine matrices as giant containers with shelves, where each shelf holds specific values. A matrix is an array of numbers arranged in rows and columns that can be used to represent a set of equations or to perform various operations, including addition, subtraction, and multiplication. The power of matrices lies in their ability to simplify complex calculations that involve multiple variables. They're like bucket brigades for numbers, passing quantities along their rows and columns to ease the burden of hefty arithmetic.

For a visual metaphor, think of a checkerboard or a spreadsheet, where each cell can store a number, and the whole structure represents a collection of these numbers. In the context of our exercise, the matrix is a tool that helps simplify the calculation process for adding several pairs of numbers in a structured way. Each 'checker spot' or cell in a matrix holds value, and these values can be manipulated according to the rules of matrix operations.

Corresponding Entries

Now let's explore the individual players on this team—corresponding entries. Each position in a matrix corresponds to a specific 'address' defined by its row and column. When two matrices engage in the tango of addition, each dancer—apologies, each entry—partners with its match by the coordinate of its 'address.' Only the numbers dancing on the same spot can be added together. This is due to the fundamental rule that matrices are combined element-wise.

For example, in our exercise, to find the sum of matrix A and matrix C, we look for a partner for the number 5 in matrix A, which is -7 in matrix C because both are in the first row and first column. They are corresponding entries. Each resulting 'dance move' leads to a new matrix with entries that are the sum of their corresponding partners.

Matrix Dimensions

Before any matrix operation can begin, it is crucial to check if the participants—the matrices—are compatible. This compatibility is determined by their dimensions. Matrix dimensions are like the measurements of a box, given by the count of rows followed by the count of columns, often written as 'rows x columns.' Two matrices must be of the same size, the same number of rows and columns, to be eligible to play the addition game.

In our example, we ensure that both matrices A and C are playing in the same league by verifying that they both measure 2x3—two rows and three columns. This matching aspect of dimensions allows for the paired addition of corresponding entries. Without this match, like trying to connect two puzzle pieces from different puzzles, the operation just wouldn't work.