Problem 24 - 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) \(3 B+2 C\) (b) \(2 H+D\)

Step-by-Step Solution

Answer

(a) \(3B + 2C = \left[ \begin{array}{rrr} 13 & -\frac{7}{2} & 15 \\ 3 & 1 & 3 \end{array} \right]\); (b) operation cannot be performed.

Step 1: Check Dimensions for 3B + 2C

Matrix addition requires matrices to have the same dimensions. Matrix \(B\) is a 2x3 matrix, and matrix \(C\) is also a 2x3 matrix. Therefore, matrices \(B\) and \(C\) have the same dimensions, and we can perform the operation \(3B + 2C\).

Step 2: Perform Scalar Multiplication on 3B

Multiply each entry of matrix \(B\) by 3: \[3B = 3 \times \left[ \begin{array}{rrr} 3 & \frac{1}{2} & 5 \ 1 & -1 & 3 \end{array} \right] = \left[ \begin{array}{rrr} 9 & \frac{3}{2} & 15 \ 3 & -3 & 9 \end{array} \right]\]

Step 3: Perform Scalar Multiplication on 2C

Multiply each entry of matrix \(C\) by 2: \[2C = 2 \times \left[ \begin{array}{rrr} 2 & -\frac{5}{2} & 0 \ 0 & 2 & -3 \end{array} \right] = \left[ \begin{array}{rrr} 4 & -5 & 0 \ 0 & 4 & -6 \end{array} \right]\]

Step 4: Add Matrices 3B and 2C

Add the corresponding elements of matrices \(3B\) and \(2C\): \[3B + 2C = \left[ \begin{array}{rrr} 9 & \frac{3}{2} & 15 \ 3 & -3 & 9 \end{array} \right] + \left[ \begin{array}{rrr} 4 & -5 & 0 \ 0 & 4 & -6 \end{array} \right] = \left[ \begin{array}{rrr} 13 & -\frac{7}{2} & 15 \ 3 & 1 & 3 \end{array} \right]\]

Step 5: Check Dimensions for 2H + D

Matrix addition requires matrices to have the same dimensions. Matrix \(H\) is a 2x2 matrix and matrix \(D\) is a 1x2 matrix. Since the dimensions differ, the operation \(2H + D\) cannot be performed.

Key Concepts

Matrix Addition

Matrix addition is like adding numbers, but instead of dealing with single values, we work with arrays of values. Each array, or matrix, is essentially a two-dimensional collection of numbers arranged in rows and columns.
However, when we add matrices together, there’s a crucial rule: the matrices must be of the same dimensions. This means the number of rows and columns in each matrix must match.

For example, in our problem, matrices \(B\) and \(C\) are both 2x3 matrices, which implies two rows and three columns each. Given these identical dimensions, we are allowed to add them by simply adding corresponding elements from each matrix. Here's how it works:

• Add the element in the first row, first column of matrix \(B\) to the element in the first row, first column of matrix \(C\).
• Repeat this for all corresponding elements, moving across rows and down columns.

Thus, when we add matrices \(3B\) and \(2C\), their result is a new 2x3 matrix whose elements are the sums of the corresponding elements from \(3B\) and \(2C\). Matrices that don't share the same dimensions, like in the case of \(2H + D\), simply can't be added.

Scalar Multiplication

Scalar multiplication involves multiplying every element of a matrix by a single number, known as the scalar. Think of it as stretching or shrinking the entire matrix by a factor, the scalar.
This operation is straightforward because you apply the scalar to each element within the matrix independently.

Consider multiplying matrix \(B\) by the scalar 3 as given in the challenge:

• Multiply each element in matrix \(B\) by 3. For instance, if an element is 2, then multiplying it by 3 gives 6 — simple!
• This results in a new matrix, \(3B\), which retains the same dimensions as the original matrix \(B\).

This process is essential not only for performing matrix addition on scaled matrices like \(3B\) and \(2C\), but also in various other mathematical and applied contexts. Note that, while the same process was applied to matrix \(C\) with scalar 2, ensuring consistent matrix dimensions simplifies further operations.

Matrix Dimensions

Matrix dimensions are a key concept in understanding matrix operations. Dimensions are defined by the number of rows and columns in a matrix, often referred to as \(m \times n\), where \(m\) is the number of rows and \(n\) the number of columns.
These dimensions dictate what operations can be performed on the matrix.

Knowing the dimensions helps determine compatibility in operations like matrix addition, where matrices must share the same dimensions. For example:

• If one matrix is 2x3, another matrix must also be 2x3 to add them.
• Conversely, any operation that involves matrices of differing dimensions may result in an operation that cannot be completed, such as \(2H + D\) in our problem, due to mismatched dimensions.

Understanding dimensions is not only essential for performing basic operations but also for more complex matrix manipulations and applications in fields like computer graphics, statistics, and signal processing.