Problem 22 - Algebra and Trigonometry Real Mathematics, Real People

Question

Evaluating an Expression Evaluate the expression. $$\left[\begin{array}{rr} 6 & 9 \\ -1 & 0 \\ 7 & 1 \end{array}\right]+\left[\begin{array}{rr} 0 & 5 \\ -2 & -1 \\ 3 & -6 \end{array}\right]+\left[\begin{array}{rr} -13 & -7 \\ 4 & -1 \\ -6 & 0 \end{array}\right]$$

Step-by-Step Solution

Answer

The answer will be the matrix \[ \begin{pmatrix} -7 & 7 \\ 1 & -2 \\ 4 & -5 \end{pmatrix} \].

Step 1: Identifying the Elements in the Same Position

For any two matrices to be added or subtracted, they must have the same order, i.e., the same number of rows and columns. Here, all the three matrices are of order 3x2 which means they have 3 rows and 2 columns. So, corresponding positions across the three matrices can be easily identified. For instance, in the first row and first column, the elements are 6, 0, and -13 in the first, second and third matrices respectively.

Step 2: Adding the Elements

In each position, add the corresponding elements. Start from the first row and first column. Therefore, (6 + 0 -13) gives -7. Do this for every position in the matrices.

Step 3: Final Matrix

Write each result in a new matrix in the corresponding position. The result will also be a matrix of order 3x2 as the order of a resultant matrix is same as the matrices being added or subtracted.

Key Concepts

Matrices Arithmetic

Understanding matrices arithmetic is crucial for manipulating and combining numerical data in a structured form. In mathematics, matrices are arrays of numbers, symbols, or expressions arranged in rows and columns, creating a rectangular grid.

Matrix addition is one of the primary operations in matrices arithmetic, allowing us to combine the information from different matrices into a single matrix. The key to successful matrix addition lies in making sure that the matrices involved have the same dimensions, meaning the same number of rows and columns.

When adding matrices, we proceed element by element. This means each entry in the resulting matrix is the sum of the corresponding entries in the matrices being added together. For example, if we have a matrix A with an element a_ij (in the ith row and jth column) and a matrix B with an element b_ij, their sum at the same position in the resulting matrix C will be c_ij = a_ij + b_ij.

It's crucial to remember that matrix addition is only possible when the matrices have the same order; otherwise, the operation is undefined. To visualize, imagine a structured block-by-block addition, much like putting together pieces of a puzzle that fit perfectly.

Evaluating Expressions with Matrices

Evaluating expressions with matrices involves performing operations such as addition, subtraction, multiplication, or scalar multiplication according to set rules. Key to this process is understanding that matrix operations must be carried out element by element and that the dimensions must be compatible.

Let's focus on addition as our expression evaluation entails this operation. When evaluating matrix expressions, always look at one element position at a time, taking the corresponding elements from each matrix and combining them according to the operation specified. Once all positions have been evaluated, the expression as a whole takes shape in the form of a new matrix.

To successfully evaluate matrix expressions, two important tips can aid any student or practitioner:

• Ensure consistency in the matrix dimensions for operations like addition and subtraction.
• Follow the operation rules precisely, such as addition and multiplication being carried out element-wise, while being aware of the special rules for matrix multiplication when it is required.

Matrix Order

The concept of matrix order is foundational to understanding matrix operations. The 'order' of a matrix is effectively a description of its size and is given by the number of rows followed by the number of columns, often expressed as 'm x n', where m is the number of rows and n is the number of columns.

The importance of matrix order is underscored when performing arithmetic operations. For matrices to be added or subtracted, for instance, they must be of the same order, as in the exercise where we saw all matrices had an order of 3x2. This prerequisite allows for the corresponding elements in each matrix to align properly so they can be combined through addition or subtraction.

When evaluating expressions with matrices, always verify that the matrices involved have the same order before proceeding. If there's a mismatch in order, operations like addition or subtraction cannot be performed. The order also affects other operations, like multiplication, where the number of columns in the first matrix must match the number of rows in the second matrix for the operation to be valid.