Problem 77 - College Algebra

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I added matrices of the same order by adding corresponding elements

Step-by-Step Solution

Answer

The statement 'I added matrices of the same order by adding corresponding elements' makes sense. Indeed, for matrices of the same order, the sum may be computed by adding their corresponding elements.

Step 1: Understanding Matrix Addition

To add two matrices, they must be of the same size, i.e., have the same number of rows and columns. The added matrix is obtained by adding corresponding elements of the two matrices. For instance, given two 2x2 matrices A = \[\begin{bmatrix} a & b \\ c & d \\ \end{bmatrix}\] and B = \[\begin{bmatrix} e & f \\ g & h \\ \end{bmatrix}\], their sum is calculated as: A + B = \[\begin{bmatrix} a+e & b+f \\ c+g & d+h \\ \end{bmatrix}\]. Therefore, the statement that 'matrices of the same order are added by adding corresponding elements' is correct.

Step 2: Application

Take Matrix A = \[\begin{bmatrix} 1 & 2 \\ 3 & 4 \\ \end{bmatrix}\] and Matrix B = \[\begin{bmatrix} 5 & 6 \\ 7 & 8 \\ \end{bmatrix}\]. If you apply the rule stated in Step 1, you will get Matrix C = A + B = \[\begin{bmatrix} 1+5 & 2+6 \\ 3+7 & 4+8 \\ \end{bmatrix}\] = \[\begin{bmatrix} 6 & 8 \\ 10 & 12 \\ \end{bmatrix}\]. This confirms that the statement makes sense.

Key Concepts

Corresponding Elements in Matrix Addition

When performing matrix addition, the focus is on pairing and adding corresponding elements from each matrix involved. But what exactly does "corresponding elements" mean? It refers to elements that exist in the same position within their respective matrices. For instance, if you are dealing with two matrices called Matrix A and Matrix B, the elements located at the first row, first column in each matrix are considered corresponding. Similarly, the elements located at the second row, second column are also corresponding.

The operation is simple:

• Add the elements in the first row, first column of Matrix A and Matrix B.
• Repeat this for each subsequent position: second row, first column, and so on.

Every paired element results in a new element in the resulting matrix of the same order. This ensures that the addition process retains the structures of the initial matrices without error. Understanding corresponding elements is vital for a flawless matrix addition process.

Understanding Matrix Order

Matrix order is a fundamental concept when dealing with matrices. It refers to the dimensions of a matrix, commonly expressed as 'rows x columns.' For example, a matrix with 2 rows and 3 columns has an order of 2x3.

This is essential information when performing operations on matrices, such as addition, because it dictates how the elements are organized. A matrix of order 3x3 will have 3 rows and 3 columns, housing 9 elements in total.

Properly comprehending the matrix order is crucial. Why? Because when two matrices are being added, they must share the same order. Each element in a position within one matrix needs to find its pair at the exact position in the other matrix - this is only possible if both matrices have the same rows and columns. Therefore, matrix order provides the structure needed for seamless addition.

Matrices of the Same Size Requirement

The rule that two matrices must be the same size to be added is simple yet important. A matrix's "size" refers directly to its order, which includes the number of rows and columns it contains. For two matrices to be compatible for addition, every single spot in one matrix must have a corresponding spot in the other.

This implies:

• The first matrix could be a 2x2, and the second must also be 2x2.
• Similarly, a 3x1 matrix can only be added to another 3x1 matrix.

The need for matrices to be of the same size ensures that each element in the first matrix can correctly pair with a counterpart element in the second. This matching arrangement simplifies computations and keeps mathematical operations consistent and true. Without equal-sized matrices, the concept of corresponding elements becomes invalid, making the addition impossible.