Problem 23 - Precalculus: Mathematics for Calculus

Question

The matrices \(A, B, C, D, E, F, G\) and \(H\) are defined as follows. $$A=\left[\begin{array}{rr}2 & -5 \\\0 & 7\end{array}\right] \quad B=\left[\begin{array}{rrr}3 &\frac{1}{2} & 5 \\\1 & -1 & 3\end{array}\right] \quad C=\left[\begin{array}{rrr}2 & -\frac{5}{2} &0 \\\0 & 2 & -3\end{array}\right]$$ $$D=\left[\begin{array}{lll}7 & 3\end{array}\right] \quad E=\left[\begin{array}{l}1 \\\2 \\\0\end{array}\right] \quad F=\left[\begin{array}{lll}1 & 0 & 0 \\\0 & 1 & 0 \\\0 & 0 & 1\end{array}\right]$$ $$G=\left[\begin{array}{rrr}5 & -3 & 10 \\\6 & 1 & 0 \\\\-5 & 2 & 2\end{array}\right] \quadH=\left[\begin{array}{rr}3 & 1 \\\2 & -1\end{array}\right]$$ 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 Matrix Dimensions for B+C

Matrix addition requires both matrices to have the same dimensions. Matrix \( B \) is a \( 2 \times 3 \) matrix, as it has 2 rows and 3 columns. Matrix \( C \) is also a \( 2 \times 3 \) matrix. Since both matrices have identical dimensions, we can add them together.

Step 2: Add Matrices B and C Element-wise

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

Step 3: Check Matrix Dimensions for B+F

Matrix \( B \) is a \( 2 \times 3 \) matrix, and matrix \( F \) is a \( 3 \times 3 \) matrix. Since they do not have the same dimensions, matrix addition is not possible.

Key Concepts

Matrix Addition

Matrix addition is a fundamental operation in linear algebra. It involves adding two matrices by adding their corresponding elements together. This operation can only be performed when both matrices have identical dimensions. If the dimensions differ, matrix addition cannot be carried out.

Matrix addition is straightforward when you follow these steps:

• Verify that both matrices have the same number of rows and columns.
• Add each corresponding element from both matrices together to form a new matrix.

For example, if you have two matrices of dimensions \(2 \times 3\), you can add them to get another matrix of the same dimensions. This process is useful in many areas of mathematics and applied sciences, as it allows for easy combination and manipulation of data.

Matrix Dimensions

Understanding matrix dimensions is crucial for performing any matrix operations, including addition. The dimensions of a matrix are described as \(m \times n\), where \(m\) is the number of rows and \(n\) is the number of columns. These dimensions dictate whether or not certain operations can be performed.

When dealing with matrices:

• Check the number of rows and columns in each matrix before attempting addition.
• Ensure matrices share the same dimensions to add them successfully.
• Remember that different operations may have different requirements for dimension compatibility, like multiplication.

For instance, matrix \(B\) has dimensions \(2 \times 3\) and can only be added to another matrix with the same \(2 \times 3\) dimensions, such as matrix \(C\). Recognizing and matching dimensions is pivotal in carrying out accurate mathematical operations and avoiding errors.

Element-wise Addition

Element-wise addition is a specific technique within matrix addition. It involves directly adding the elements of two matrices that occupy the same position, resulting in a new matrix of the same dimensions. This method is used exclusively for matrices that are of the same size.

To perform element-wise addition:

• Identify the corresponding elements in each matrix.
• Add these elements together.
• Store the sum in the resulting matrix at the same position.

For example, for matrices \(B\) and \(C\), which are both \(2 \times 3\), the element-wise addition is:\[B + C = \begin{bmatrix} 3+2 & \frac{1}{2} + \left(-\frac{5}{2}\right) & 5+0 \ 1+0 & -1+2 & 3+\left(-3\right) \end{bmatrix} = \begin{bmatrix} 5 & -2 & 5 \ 1 & 1 & 0 \end{bmatrix}\]This approach simplifies the process of combining matrices by treating each element in isolation, ensuring that the matrix structure remains consistent and true to its original form.