Problem 17 - Precalculus: Mathematical for Calculus

Question

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

Step-by-Step Solution

Answer

Matrix addition is possible; the result is \(\begin{bmatrix} 5 & -2 & 5 \\ 1 & 1 & 0 \end{bmatrix}\).

Step 1: Understand Matrix Addition

Matrix addition is possible only when both matrices have the same dimensions (i.e., same number of rows and columns). For this exercise, we need to determine if matrices \(B\) and \(C\) have the same dimensions.

Step 2: Check Dimensions of Matrix B

Matrix \(B\) is a 2x3 matrix: it has 2 rows and 3 columns. This means any matrix to be added to \(B\) must also be a 2x3 matrix.

Step 3: Check Dimensions of Matrix C

Matrix \(C\) is also a 2x3 matrix with 2 rows and 3 columns, just like matrix \(B\). This implies that matrix addition is possible between \(B\) and \(C\).

Step 4: Perform Matrix Addition

Since both matrices \(B\) and \(C\) are 2x3 matrices, we can add them by adding the corresponding elements. Thus: \[B + C = \begin{bmatrix}3 & \frac{1}{2} & 5 \1 & -1 & 3 \end{bmatrix} +\begin{bmatrix}2 & -\frac{5}{2} & 0 \0 & 2 & -3\end{bmatrix}\]The resulting matrix will be: \[= \begin{bmatrix}3+2 & \frac{1}{2} + (-\frac{5}{2}) & 5+0 \1+0 & -1+2 & 3+(-3)\end{bmatrix} = \begin{bmatrix}5 & -2 & 5 \1 & 1 & 0\end{bmatrix}\]

Step 5: Final Result

The resulting matrix from \(B+C\) is \[ \begin{bmatrix} 5 & -2 & 5 \ 1 & 1 & 0 \end{bmatrix} \].

Key Concepts

Matrix Dimensions

The dimension of a matrix is a fundamental concept in matrix algebra. It is defined by two numbers: the number of rows and the number of columns. For example, a matrix with 2 rows and 3 columns is described as a 2x3 matrix. These dimensions are crucial when performing operations like addition, subtraction, or multiplication as they determine if certain operations are feasible.

When working with matrices, ensuring that the dimensions align for specific operations is key. For instance, matrix addition requires both matrices to have exactly the same dimensions. In our example, to add matrices B and C together, both must have the same number of rows and columns, which they do, as they are both 2x3 matrices. This makes the addition operation possible.

Matrix Operations

Matrix operations are the procedures you perform on matrices to manipulate them or extract information. The most common matrix operations include addition, subtraction, and multiplication.

Addition

For matrix addition, the matrices must have identical dimensions. Once confirmed, you add corresponding elements, meaning the element in the first row, first column of one matrix is added to the first row, first column of the other matrix, and so on. This operation simplifies to just adding numbers, but across each matrix's respective position.

Multiplication

Matrix multiplication is a bit more complex than addition. Here, the number of columns in the first matrix must equal the number of rows in the second for the operation to be valid. Each element in the resulting matrix is calculated by taking a row from the first matrix and a column from the second, multiplying their respective elements, and then summing those products.

Knowing these basic operations helps in solving many mathematical problems involving matrices. In our exercise, the addition of the matrices B and C illustrates the simplicity and beauty of matrix addition once the dimensions are verified.

Matrices

Matrices are an array of numbers, symbols, or expressions arranged in rows and columns. They are a powerful tool for mathematical calculations and are used in various fields like economics, engineering, physics, computer graphics, and more.

Structure and Elements

Each number within a matrix is termed an element. The position of each element is noted by its row and column number, which defines the matrix's ability to perform certain operations. Understanding this layout is crucial for completing operations correctly.

Types and Uses

Matrices can come in various forms, such as row matrices, column matrices, square matrices, and identity matrices. Each type serves specific purposes in different applications. For example, identity matrices are equivalent to the number 1 in scalar math and are used in operations like finding inverse matrices.

Overall, matrices provide a structured and efficient way to handle and manipulate large datasets and are essential in modern computational techniques. By practicing with different operations and types, like those provided in matrix algebra, one can gain a deeper understanding and fluency in applying them to solve practical problems.