Introduction

The order of a matrix is a fundamental concept in linear algebra, critical for understanding how matrices can be multiplied and what the order of the resultant matrix would be. In the context of the expression 2A 5B 7C - 8D, we need to determine the order of the resultant matrix after performing the given operations. This article will guide you through the process, highlighting the importance of matrix dimensions and providing a step-by-step approach to solving such problems.

Steps to Find the Order of the Resultant Matrix

Identify the Orders of Each Matrix

To begin, it's essential to understand the dimensions of the matrices involved. Let's denote:

The order of matrix (A) as (m times n). The order of matrix (B) as (m times n). The order of matrix (C) as (n times p). The order of matrix (D) as (n times p).

Calculate the Order of the Resultant Matrices

Given the orders mentioned, we can calculate the resultant orders of the following expressions:

The expression (2A 5B): Since (2A) and (5B) have the same dimensions, the resultant matrix will also have the order (m times n). The expression (7C - 8D): Since both (7C) and (8D) have the same dimensions, the resultant matrix will also have the order (n times p).

Multiply the Resultant Matrices

For matrix multiplication to be valid, the number of columns in the first matrix must match the number of rows in the second matrix. In our case, we need to multiply the following matrices:

The first matrix from the expression is (2A 5B) with the order (m times n). The second matrix from the expression is (7C - 8D) with the order (n times p).

Since the number of columns in the first matrix ((n)) is equal to the number of rows in the second matrix ((n)), the multiplication is valid.

Determine the Order of the Resultant Matrix

When multiplying two matrices (P) and (Q) with orders (m times n) and (n times p) respectively, the resultant matrix (R) will have the order (m times p). Therefore, the resultant matrix from multiplying (2A 5B) by (7C - 8D) will have the order (m times p).

Summary

The order of the resultant matrix from the expression (2A 5B 7C - 8D) is (m times p) where (m) is the number of rows in matrices (A) and (B), and (p) is the number of columns in matrices (C) and (D).

This solution is based on the principle that matrix addition and scalar multiplication do not change the order of the matrices, and matrix multiplication requires the number of columns in the first matrix to match the number of rows in the second matrix.

Additional Insights

Let's consider the case where (p), (q), and (r) are any three positive integers. If both matrices (A) and (B) are of order (p times q) and both matrices (C) and (D) are of order (q times r), then the expressions (2A 5B) and (7C - 8D) both exist as matrices of order (p times q) and (q times r) respectively. In this scenario, the expression (2A 5B 7C - 8D) is valid and the resulting matrix will be of order (p times r).

Conclusion

Understanding the order of matrices is crucial for performing mathematical operations on them. The order of matrices and the rules of matrix operations are deeply interconnected, and mastering them will enhance your ability to work with matrices in various mathematical and scientific applications.