Problem 68 - Algebra and Trigonometry

Question

How are matrices added?

Step-by-Step Solution

Answer

Given two matrices of the same dimension, they are added by creating a new matrix where each element is the sum of the corresponding elements from the two original matrices.

Step 1: Identify the Matrices

Identify the two matrices that you are supposed to add. Let's take two 2x2 matrices A = \[\[a_{11}, a_{12}\], \[a_{21}, a_{22}\]\] and B = \[\[b_{11}, b_{12}\], \[b_{21}, b_{22}\]\] for example.

Step 2: Confirming Compatibility

Ensure the two matrices have the same dimensions, that is, they have the same number of rows and columns. In this case, both A and B are 2x2 matrices.

Step 3: Add Corresponding Elements

Add the respective elements from each matrix to form a new matrix which is the sum of the two matrices. The resulting matrix C = A + B will also be a 2x2 matrix where C = \[\[a_{11}+b_{11}, a_{12}+b_{12}\], \[a_{21}+b_{21}, a_{22}+b_{22}\]\].

Key Concepts

Matrices

Matrices are fundamental concepts in mathematics and are essentially rectangular arrays of numbers, symbols, or expressions arranged in rows and columns. They are used to represent and solve systems of linear equations, among various other applications. You can think of a matrix as a structured way of organizing a set of data, similar to a spreadsheet.

• Matrices are denoted by upper-case letters such as A or B.
• Each item within a matrix is called an "element," which we identify by its position.

Alongside being practical tools in mathematical computations, matrices can represent data in formats that are easy to manipulate, analyze, and perform operations on, such as matrix addition or multiplication.

Matrix Dimensions

The dimensions of a matrix are a way of describing its size. They tell you the number of rows and columns a matrix has. Dimensions are given in the format \(m \times n\), where \(m\) is the number of rows, and \(n\) is the number of columns.

• A 2x2 matrix has 2 rows and 2 columns.
• Understanding dimensions is essential for matrix operations, as operations like addition require matrices to have the same dimensions.

When you read or write matrices, always consider their dimensions first to ensure mathematical operations are possible.

Matrix Elements

Matrix elements are the individual values or entries contained within a matrix. Each element is generally represented by a lowercase letter with two subscripts that indicate its position within the matrix.

• The first subscript usually denotes the row number.
• The second subscript denotes the column number.

For example, in a matrix A represented as \(A = \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix}\), \(a_{11}\) is the element in the first row and first column. This system of labeling helps precisely locate and reference any specific element within a matrix.

Addition of Matrices

Adding matrices involves combining two matrices of the same dimensions by adding their corresponding elements. This operation results in a new matrix of the same dimensions.

• To add two matrices, say A and B, both must have identical dimensions, such as 2x2.
• The sum matrix, C, involves adding each element in matrix A to the corresponding element in matrix B.

For instance, given \(A = \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix}\) and \(B = \begin{bmatrix} b_{11} & b_{12} \ b_{21} & b_{22} \end{bmatrix}\), the result of \(C = A + B\) is \(C = \begin{bmatrix} a_{11}+b_{11} & a_{12}+b_{12} \ a_{21}+b_{21} & a_{22}+b_{22} \end{bmatrix}\). This process highlights the simplicity and efficiency of matrix addition when the dimensions are compatible.