Problem 93 - Algebra and Trigonometry Real Mathematics, Real People

Question

True or False? Determine whether the statement is true or false. Justify your answer. Two matrices can be added only when they have the same dimension.

Step-by-Step Solution

Answer

The statement 'Two matrices can be added only when they have the same dimension' is True. This is because in matrix addition, elements at corresponding positions in the matrices are added which can only be done if both matrices have the same dimensions.

Step 1: Understanding matrix addition

Matrix addition requires that the two matrices should have the same dimension. In other words, they should have same number of rows and same number of columns. Equal-sized matrices are necessary because in matrix addition, corresponding elements are added together. If one matrix has, for example, extra rows or columns, there won't be corresponding elements to add in the other matrix.

Step 2: Compare statement with the rule of matrix addition

The statement given in the problem is 'Two matrices can be added only when they have the same dimension'. This is exactly same as the rule of matrix addition meaning if two matrices do not have the same number of rows and columns, they cannot be added together.

Step 3: Make a decision based on the established facts

After comparing the rule of matrix addition with the statement, we can determine that the statement is true. Hence, it is correct to say that 'Two matrices can be added only when they have the same dimension'.

Key Concepts

Matrix Dimensions

In the world of matrices, dimensions are crucial. Every matrix has rows and columns, and they define its structure. For instance, if a matrix has 3 rows and 2 columns, it’s said to have dimensions of 3x2. The number of rows always comes first, followed by the number of columns.
When it comes to matrix operations like addition or subtraction, matching dimensions are a requirement. Think of dimensions like a grid layout that two matrices must share to work cooperatively. Only matrices with exactly the same number of rows and columns can be seamlessly manipulated together.
Without matching dimensions, certain calculations, like adding matrices, simply aren’t possible because there isn’t a one-to-one match of their parts, or 'corresponding elements', which we'll discuss further.

Matrix Operations

Matrix operations involve a number of transformative procedures such as addition, subtraction, and multiplication. Each operation has its own specific rules that must be followed to ensure correct calculations. For addition and subtraction, only matrices of the same dimensions can be directly altered together.
When adding matrices, each element in one matrix is paired with its corresponding counterpart in the other. The two are summed, creating a new matrix where each element represents a sum of these pairs.
Here's a quick look at how simple matrix addition works:

• Identify matching elements: The first element of the first row from each matrix are added together.
• Add these elements together and place the sum in a new matrix at the same position.
• Repeat for every element in the matrices.

This step-by-step process goes hand-in-hand with maintaining identical dimensions, as it ensures each element has a partner to transform with.

Corresponding Elements

To understand matrix addition deeply, focusing on corresponding elements is important. Corresponding elements are simply the 'matched' elements in two matrices that sit in the same row and column. They are the core of what makes dimension-matching crucial.
When two matrices have the same dimensions, each element in the first matrix has a direct counterpart in the second matrix—like having two stacks of coins with the same number on each stack. Each top coin is added to the coin directly opposite it.
Corresponding elements always "live" in the same position across both grids. For example, in two 3x3 matrices, the element in the second row and third column of the first matrix will be paired with the element in the second row and third column of the second matrix.
This method ensures all parts of both matrices contribute to forming answers that are both accurate and reliable.