Problem 17 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The resulting matrix is \( \left[\begin{array}{rrr}5 & -2 & 5 \\ 1 & 1 & 0\end{array}\right] \).

Step 1: Check Compatibility of Matrices for Addition

Before adding two matrices, they must have the same dimensions. Matrix \( B \) is a \( 2 \times 3 \) matrix and matrix \( C \) is also a \( 2 \times 3 \) matrix. Since they have the same number of rows and columns, they can be added together.

Step 2: Add Corresponding Elements

To add matrices \( B \) and \( C \), we add each corresponding element: \[ B + C = \left[\begin{array}{rrr}{3+2} & {\frac{1}{2} + (-\frac{5}{2})} & {5+0} \ {1+0} & {-1+2} & {3-3}\end{array}\right] \]

Step 3: Simplify the Resultant Matrix

Simplify the elements of the resultant matrix:\[ B + C = \left[\begin{array}{rrr}{3+2} & {\frac{1}{2} - \frac{5}{2}} & {5} \ {1} & {1} & {0}\end{array}\right] = \left[\begin{array}{rrr}{5} & {-2} & {5} \ {1} & {1} & {0}\end{array}\right] \]

Key Concepts

Matrix Dimensions

When working with matrices, understanding their dimensions is crucial. Matrix dimensions are expressed in terms of their rows and columns. For instance, a matrix with 2 rows and 3 columns is said to be a \(2 \times 3\) matrix. Knowing the dimensions is essential because it determines whether certain operations, like addition or multiplication, can be performed.

• Rows refer to the horizontal arrangement of numbers.
• Columns are the vertical arrangement of numbers.
• Dimensions ensure compatibility for various matrix operations.

In matrix addition, it is vital that both matrices share the same dimensions. This means they must have an equal number of rows and columns. If two matrices have different dimensions, they cannot be directly added or subtracted. Knowing this can save you time when solving matrix-related problems.

Matrix Operations

Matrix operations encompass a variety of methods for manipulating matrices, including addition, subtraction, multiplication, and more. Each operation has specific rules that must be followed to ensure proper calculation.

Matrix Addition

Matrix addition involves adding two matrices element by element. This operation requires both matrices to have the same dimensions, ensuring there's a one-to-one correspondence between their elements. To perform addition:

• Align the matrices based on their dimensions.
• Add corresponding elements (i.e., elements in the same position in each matrix).
• Formula: \( (A + B)_{ij} = A_{ij} + B_{ij} \) where \(i\) and \(j\) are the row and column indices, respectively.

Matrix operations can be quite powerful, providing a means to solve complex algebraic systems and perform a multitude of computational tasks.

Algebraic Operations

Algebraic operations with matrices mirror many of the rules from basic algebra, but their unique properties and rules must be understood. When adding matrices, the operation is straightforward but requires precision.

Checking for Compatibility

As highlighted in matrix operations, compatibility is key in algebraic operations. To ensure matrices can be added, always verify their dimensions. For example:

• Matrix \( B \) is \(2 \times 3\).
• Matrix \( C \) is \(2 \times 3\).
• Since their dimensions match, the sum \( B + C \) is possible.

Simplification

After performing the algebraic operation of addition, the resultant matrix must be simplified. Simplifying ensures each element is reduced to its most straightforward form. This could involve performing basic arithmetic on fractions or whole numbers. Through these steps, matrices provide a methodical approach to solving complex problems both numerically and visually, enhancing your understanding of algebraic relationships.