Problem 53 - Algebra and Trigonometry Real Mathematics, Real People

Question

Operations with Matrices Use the matrix capabilities of a graphing utility to evaluate the expression. $$\left[\begin{array}{rrr} 0 & 2 & -2 \\ 4 & 1 & 2 \end{array}\right]\left(\left[\begin{array}{rr} 4 & 0 \\ 0 & -1 \\ -1 & 2 \end{array}\right]+\left[\begin{array}{rr} -2 & 3 \\ -3 & 5 \\ 0 & -3 \end{array}\right]\right)$$

Step-by-Step Solution

Answer

The result of the operation is the matrix \( \left[\begin{array}{rr} -8 & 6 \ 4 & 17 \end{array}\right] \)

Step 1: Adding Matrices

Begin by adding together the two matrices inside the parentheses which gives a resultant matrix R. \[ R = \left[\begin{array}{rr} 4 & 0 \ 0 & -1 \ -1 & 2 \end{array}\right] + \left[\begin{array}{rr} -2 & 3 \ -3 & 5 \ 0 & -3 \end{array}\right] = \left[\begin{array}{rr} 2 & 3 \ -3 & 4 \ -1 & -1 \end{array}\right] \]

Step 2: Multiplying Matrices

Next, multiply the resulting matrix R by the matrix outside the parentheses. The resulting product is the final answer: \[ \left[\begin{array}{rrr} 0 & 2 & -2 \ 4 & 1 & 2 \end{array}\right] \times \left[\begin{array}{rr} 2 & 3 \ -3 & 4 \ -1 & -1 \end{array}\right] = \left[\begin{array}{rr} -8 & 6 \ 4 & 17 \end{array}\right] \]

Key Concepts

Matrix Addition

Matrix addition is a fundamental operation allowing us to combine matrices by adding corresponding elements. To perform matrix addition, you must ensure that the matrices have the same dimensions. This means they must have the same number of rows and columns.
When adding matrices, simply add each element in the first matrix to the corresponding element in the second matrix. For example, adding two matrices \( A \) and \( B \) results in a new matrix \( C \), where each element \( c_{ij} \) is obtained by calculating \( a_{ij} + b_{ij} \).

• Key Rule: Only matrices with identical dimensions can be added.
• Element-wise Addition: Each element of the resulting matrix is the sum of the corresponding elements of the given matrices.

In the original exercise, matrix addition was performed inside parentheses, resulting in a new matrix ready for further operations.

Matrix Multiplication

Matrix multiplication is a bit more complex than addition but is a crucial operation in matrix algebra. To multiply matrices, the number of columns in the first matrix must be equal to the number of rows in the second matrix.
The product of two matrices involves taking the dot product of rows from the first matrix with columns from the second matrix. Each element of the resulting matrix is derived from these products.

• Dimension Requirement: If matrix \( A \) is an \( m \times n \) matrix and matrix \( B \) is an \( n \times p \) matrix, the resulting product will be an \( m \times p \) matrix.
• Calculating Elements: Each element is calculated by summing the products of the corresponding elements from the rows and columns.

In the exercise, the product of the two matrices post-addition yielded the final answer.

Graphing Utility

A graphing utility is a tool that can simplify computations involving matrices, especially when dealing with large matrices or performing numerous operations. This software, typically found in graphing calculators or computer programs, helps visualize and calculate matrix operations efficiently.
Using a graphing utility, you can:

• Handle extensive matrix computations with ease.
• Verify manual calculations by cross-checking the results.
• Rapidly perform entire sequences of operations, such as addition followed by multiplication.

For students, leveraging a graphing utility can lead to increased understanding and accuracy, as it reduces errors that might occur when performing complex manual computations.

Evaluating Expressions

Evaluating matrix expressions involves performing a series of operations in the correct order, much like evaluating algebraic expressions. It's essential to follow the order of operations, which can include operations like addition, subtraction, and multiplication of matrices.
When evaluating a matrix expression, consider the following:

• Order of Operations: Follow the PEMDAS rule – Parentheses first, then Exponents (though not typically in basic matrix operations), followed by Multiplication and Division, and finally Addition and Subtraction.
• Computational Accuracy: Ensure each step is completed correctly to arrive at the accurate final result.

In our given exercise, we first perform matrix addition (inside the parentheses) before moving on to matrix multiplication, ensuring that we respect the natural order of operations to achieve the correct solution.