Problem 21 - Precalculus Mathematics for Calculus

Question

The matrices \(A, B, C, D, E, F, G\) and \(H\) are defined as follows. $$\begin{aligned} &A=\left[\begin{array}{rr} 2 & -5 \\ 0 & 7 \end{array}\right] \quad B=\left[\begin{array}{rrr} 3 & \frac{1}{2} & 5 \\ 1 & -1 & 3 \end{array}\right] \quad C=\left[\begin{array}{rrr} 2 & -\frac{5}{2} & 0 \\ 0 & 2 & -3 \end{array}\right]\\\ &D=\left[\begin{array}{llll} 7 & 3 \end{array}\right] \quad E=\left[\begin{array}{l} 1 \\ 2 \\ 0 \end{array}\right] \quad F=\left[\begin{array}{lll} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]\\\ &G=\left[\begin{array}{rrr} 5 & -3 & 10 \\ 6 & 1 & 0 \\ -5 & 2 & 2 \end{array}\right] \quad H=\left[\begin{array}{rr} 3 & 1 \\ 2 & -1 \end{array}\right] \end{aligned}$$ Carry out the indicated algebraic operation, or explain why it cannot be performed. (a) \(B+C\) (b) \(B+F\)

Step-by-Step Solution

Answer

(a) \(B+C = \begin{bmatrix} 5 & -2 & 5 \\ 1 & 1 & 0 \end{bmatrix}\). (b) \(B+F\) cannot be performed.

Step 1: Check Dimensions for Addition of Matrices

Matrix addition requires both matrices to have the same dimensions. Matrix \(B\) is \(2 \times 3\) and matrix \(C\) is also \(2 \times 3\). Since they have the same dimensions, the operation \(B + C\) can be performed.

Step 2: Perform Addition of Matrices B and C

Add corresponding elements of matrices \(B\) and \(C\):\[B+C = \begin{bmatrix}3 + 2 & \frac{1}{2} + \left(-\frac{5}{2}\right) & 5 + 0 \1 + 0 & -1 + 2 & 3 + \left(-3\right)\end{bmatrix} = \begin{bmatrix}5 & -2 & 5 \1 & 1 & 0\end{bmatrix}\]

Step 3: Check Dimensions for Addition of Matrices B and F

Matrix \(B\) has dimensions \(2 \times 3\), whereas matrix \(F\) is \(3 \times 3\). Since their dimensions do not match, the operation \(B + F\) cannot be performed.

Key Concepts

Understanding Matrix Dimensions

When discussing matrices, understanding their dimensions is crucial.
The dimensions of a matrix refer to the number of rows and columns it contains. It is usually expressed as "rows by columns" (\(m \times n\)). For example:

• Matrix A: \(2 \times 2\) - 2 rows and 2 columns
• Matrix B: \(2 \times 3\) - 2 rows and 3 columns
• Matrix F: \(3 \times 3\) - 3 rows and 3 columns

In matrix operations, having the correct dimensions is necessary especially for certain operations like addition and subtraction. This plays into whether operations like matrix addition can even occur between two matrices.
For instance, both matrices B and C need to have the same dimensions of \(2 \times 3\) for \(B + C\) to proceed. However, for B and F, differing dimensions \((2 \times 3) \, \text{vs} \, (3 \times 3)\) prevent the addition operation. This highlights why checking dimensions is often the first step in matrix operations.

Basics of Matrix Operations

Matrix operations form the foundation for many applications in mathematics, computer sciences, and engineering.
The operations include addition, subtraction, multiplication, and scalar multiplication to name a few. In this exercise, we focus on addition.
Matrix addition is straightforward but requires that both matrices involved have the same dimensions. Here's how it works:

• Add corresponding elements in the two matrices.
• If matrix B is \(2 \times 3\), it must add to another \(2 \times 3\) matrix, like matrix C.
• The result will be a new matrix with identical dimensions \(2 \times 3\).

This simple rule ensures consistency in the resulting matrix's dimensions.
An example is subtle in this exercise, where matrices B and C have equal dimensions allowing their sum \(B + C\). Each element of B is added to the corresponding element in C, highlighting how dimensions dictate the possibility of the operation in matrix algebra.

Exploring Matrix Algebra

Matrix algebra is a potent mathematical tool with varied applications.
It allows solving systems of equations, transforming geometric data, or even understanding complex networks in computer science.
Matrix algebra includes a set of rules and operations used to manipulate matrices. Consider this:

• Scalar multiplication where each element is multiplied by a scalar.
• Matrix multiplication which combines transformations but requires compatible dimensions.
• And inverses which, if applicable, help solve linear systems.

However, not all operations can be performed on every matrix.
Addition, specifically, is both simple yet restricted by dimensional constraints as demonstrated in the earlier sections.
This step-by-step approach showcases how fundamental understanding of these algebraic operations allows one to perform advanced calculations. The exercise involving B and C, and why B + F is not possible, underscores not just the practical aspect but also the logical framework within which matrix algebra operates.