Question

Tell whether the matrices can be added. $$ \left[\begin{array}{rr} 4 & -1 \\ 7 & 5 \end{array}\right],\left[\begin{array}{rr} 2 & -2 \\ 5 & -6 \end{array}\right] $$

Short answer

Yes, the given matrices can be added as they both have same dimensions.

Step-by-step solution

Identify the dimensions of the matrices

Looking at both matrices, each one is a 2 by 2 matrix, containing 2 rows and 2 columns.

Compare the dimensions of both matrices

The dimensions of the first matrix are compared with the dimensions of the second matrix.

Determine if the matrices have the same dimensions

Since both matrices have 2 rows and 2 columns, their dimensions are identical. Therefore, these matrices can be added together as per the definition of matrix addition.

Key concepts

Matrices Dimensions

Understanding the dimensions of a matrix is crucial in the realm of matrix addition. A matrix's dimensions are expressed in terms of the number of rows and columns it contains, denoted as 'rows x columns'. For example, a matrix with 3 rows and 4 columns is described as a 3x4 matrix.

In the context of our exercise, we examined two matrices, each being 2x2 matrices. This means both matrices have 2 rows and 2 columns. When adding matrices, one of the primary requirements is that they must have the same dimensions. In other words, if one matrix is a 3x4, the other must also be a 3x4 to carry out the addition. If they don’t match, the operation isn't defined and can't be performed.

The dimensional compatibility assures that each element in the first matrix has a corresponding element in the second matrix at the same position. This allows for a direct element-wise addition across the matrices.

Matrices Arithmetic

Moving beyond the dimensions, let's talk about matrices arithmetic. While there are several operations we can perform on matrices, such as multiplication and transpose, we are particularly interested here in matrix addition.

Matrices arithmetic follows specific rules that differentiate it from the arithmetic of real numbers. In matrix addition, corresponding elements of each matrix are added together. For example, the element in the first row and first column of the first matrix is added to the element in the first row and first column of the second matrix, and so forth for the remaining elements.

Considering our exercise, we add the two 2x2 matrices by adding their corresponding elements: The '4' from the first matrix with the '2' from the second matrix, '-1' with '-2', '7' with '5', and '5' with '-6'. This element-by-element approach is the essence of matrix addition, making matrices a unique algebraic structure compared to other number systems.

Algebraic Structures

Finally, understanding algebraic structures is fundamental in appreciating the operations we perform on matrices. An algebraic structure consists of a set equipped with one or more operations that combine members of the set in a specific way. For matrices, the operations are not just addition; there's also multiplication, scalar multiplication, and transpose, among others.

Each operation on matrices adheres to particular rules that often mirror those in arithmetic but can differ significantly in properties. For instance, matrix multiplication is not commutative, meaning that the order in which matrices are multiplied does matter, which isn't the case in regular number multiplication.

In our exercise and its solution, we come across matrices as an example of an algebraic structure that is compatible for addition due to the size and shape of the sets being precise matches. This opens the door to further exploration of matrices and the various complex operations that can be applied, all of which are cornerstone concepts in higher mathematics, computer science, physics, and engineering.