Problem 1 - Algebra and Trigonometry

Question

We can add (or subtract) two matrices only if they have the same _____

Step-by-Step Solution

Answer

Matrices can be added or subtracted only if they have the same dimensions.

Step 1: Understand Matrix Dimensions

A matrix is defined by the number of its rows and columns. For example, a matrix with 3 rows and 2 columns is referred to as a 3x2 matrix.

Step 2: Identify Requirement for Addition/Subtraction

Matrices can only be added or subtracted if they have identical dimensions, meaning they must have the same number of rows and columns.

Step 3: Formalize the Condition

If matrix A is an m×n matrix and matrix B is also an m×n matrix, then A and B can be added or subtracted because they share the same dimensions.

Step 4: Apply the Condition to The Question

To fill in the blank, we conclude that matrices can be added or subtracted only if “they have the same dimensions”.

Key Concepts

Matrix Dimensions

A matrix is essentially a rectangular array of numbers arranged in rows and columns. Understanding matrix dimensions is crucial because it dictates how we perform operations like addition and subtraction. To describe the size of a matrix, we use two numbers:

• The number of rows
• The number of columns

These numbers are written as m×n, where m represents the number of rows and n represents the number of columns. For instance, a 3x2 matrix has 3 rows and 2 columns.
Knowing the dimensions is vital because only matrices of the same dimensions can undergo addition or subtraction. This means both matrices must have identical numbers of rows and columns. Thus, verifying matrix dimensions before any operation is a fundamental step in ensuring the operation is valid.

Matrix Addition

Matrix addition is a straightforward operation when you grasp the requirement of identical dimensions. Adding matrices involves adding their corresponding elements. Imagine two matrices, A and B, each with dimensions m×n. For the sum matrix C = A + B:

• The element at the first row and first column of C, denoted as c_{11}, is the sum of elements a_{11} from A and b_{11} from B.
• Similarly, c_{12} is the sum of a_{12} and b_{12}, and so on.

This element-wise addition continues for all elements across both matrices. The result is a new matrix, C, with the same dimensions as A and B.
Simple but crucial, matrix addition perfectly demonstrates why having the same dimensions is necessary: it ensures that each pair of elements to be added corresponds directly.

Matrix Subtraction

Much like matrix addition, matrix subtraction also requires that both matrices have the same dimensions. This operation, denoted as A - B, follows the same principle of element-wise operation, but instead, you subtract corresponding elements.

• Take the element a_{11} in matrix A and b_{11} in matrix B. The resulting element c_{11} in matrix C = A - B will be a_{11} - b_{11}.
• This process continues for every individual element; thus, c_{12} is a_{12} - b_{12}, c_{21} is a_{21} - b_{21}, etc.

Each resultant element of the new matrix retains the position as in the original matrices.
By ensuring identical dimensions in both matrices, subtractions are easily manageable and logical, providing consistency across corresponding elements. The end product, a new matrix, shares the dimensions with the original matrices being subtracted.