Problem 53 - College Algebra with Modeling and Visualization

Question

Properties of Matrices Use a graphing calculator to evaluate the expression with the given matrices \(A, B,\) and \(C .\) Compare your answers for parts (a) and (b). Then interpret the results. $$A=\left[\begin{array}{rrr}2 & -1 & 3 \\\1 & 3 & -5 \\\0 & -2 & 1\end{array}\right], B=\left[\begin{array}{rrr}6 & 2 & 7 \\\3 & -4 & -5 \\\7 & 1 & 0\end{array}\right]$$ $$C=\left[\begin{array}{lll}1 & 4 & -3 \\\8 & 1 & -1 \\\4 & 6 & -2\end{array}\right]$$ (a) \(A(B+C)\) (b) \(A B+A C\)

Step-by-Step Solution

Answer

Matrices A(B+C) and AB+AC should be equal by the distributive property of matrix multiplication.

Step 1: Calculate Matrix B+C

First, we need to determine the matrix \(B + C\). Using matrix addition, we add corresponding elements from matrices \(B\) and \(C\):\[B + C = \left[\begin{array}{rrr}6+1 & 2+4 & 7-3 \3+8 & -4+1 & -5-1 \7+4 & 1+6 & 0-2\end{array}\right] =\left[\begin{array}{rrr}7 & 6 & 4 \11 & -3 & -6 \11 & 7 & -2\end{array}\right]\]

Step 2: Compute A(B+C)

Now that we have \(B+C\), we can find \(A(B+C)\) by multiplying matrices \(A\) and \(B+C\): \[A(B+C) = \left[\begin{array}{rrr}2 & -1 & 3 \1 & 3 & -5 \0 & -2 & 1 \end{array}\right] \times \left[\begin{array}{rrr}7 & 6 & 4 \11 & -3 & -6 \11 & 7 & -2\end{array}\right] \]Calculating each element, we have:- First row: \((2 \times 7) + (-1 \times 11) + (3 \times 11), (2 \times 6) + (-1 \times -3) + (3 \times 7), (2 \times 4) + (-1 \times -6) + (3 \times -2)\)- Second row: \((1 \times 7) + (3 \times 11) + (-5 \times 11), (1 \times 6) + (3 \times -3) + (-5 \times 7), (1 \times 4) + (3 \times -6) + (-5 \times -2)\)- Third row: \((0 \times 7) + (-2 \times 11) + (1 \times 11), (0 \times 6) + (-2 \times -3) + (1 \times 7), (0 \times 4) + (-2 \times -6) + (1 \times -2)\)Calculate these values to obtain the resulting matrix.

Step 3: Compute Product AB

Next, calculate \(AB\) by multiplying matrices \(A\) and \(B\):\[AB = \left[\begin{array}{rrr}2 & -1 & 3 \1 & 3 & -5 \0 & -2 & 1 \end{array}\right] \times \left[\begin{array}{rrr}6 & 2 & 7 \3 & -4 & -5 \7 & 1 & 0\end{array}\right] \]Calculate each element:- First row: \((2 \times 6) + (-1 \times 3) + (3 \times 7), (2 \times 2) + (-1 \times -4) + (3 \times 1), (2 \times 7) + (-1 \times -5) + (3 \times 0)\)- Second row: \((1 \times 6) + (3 \times 3) + (-5 \times 7), (1 \times 2) + (3 \times -4) + (-5 \times 1), (1 \times 7) + (3 \times -5) + (-5 \times 0)\)- Third row: \((0 \times 6) + (-2 \times 3) + (1 \times 7), (0 \times 2) + (-2 \times -4) + (1 \times 1), (0 \times 7) + (-2 \times -5) + (1 \times 0)\) Perform these calculations to get the resulting matrix.

Step 4: Compute Product AC

Now, find the product \(AC\):\[AC = \left[\begin{array}{rrr}2 & -1 & 3 \1 & 3 & -5 \0 & -2 & 1 \end{array}\right] \times \left[\begin{array}{lll}1 & 4 & -3 \8 & 1 & -1 \4 & 6 & -2\end{array}\right] \]Calculate each element:- First row: \((2 \times 1) + (-1 \times 8) + (3 \times 4), (2 \times 4) + (-1 \times 1) + (3 \times 6), (2 \times -3) + (-1 \times -1) + (3 \times -2)\)- Second row: \((1 \times 1) + (3 \times 8) + (-5 \times 4), (1 \times 4) + (3 \times 1) + (-5 \times 6), (1 \times -3) + (3 \times -1) + (-5 \times -2)\)- Third row: \((0 \times 1) + (-2 \times 8) + (1 \times 4), (0 \times 4) + (-2 \times 1) + (1 \times 6), (0 \times -3) + (-2 \times -1) + (1 \times -2)\)After computing, you will obtain the resulting matrix.

Step 5: Compute AB + AC

With the results from steps 3 and 4, compute the sum \(AB + AC\). Add corresponding elements from the matrices \(AB\) and \(AC\):\[AB + AC = \left[ Result \ AB \right] + \left[ Result \ AC \right]\]Perform the matrix addition by adding each corresponding element.

Step 6: Compare Results

Now compare the matrices obtained from \(A(B+C)\) and \(AB + AC\). According to the distributive property of matrix multiplication, these two should be the same if calculations are correct.

Key Concepts

Matrix Addition

Matrix addition is a straightforward operation where two matrices of the same dimensions are combined by adding their corresponding elements. For example, if you have two matrices, say \(B\) and \(C\), each with dimensions \(3 \times 3\), you will add element-by-element:

• First row, first element of matrix \(B\) with first row, first element of matrix \(C\).
• Continue this process for all elements in the matrices.

In the exercise, we calculated matrix \(B + C\) by adding each element from matrix \(B\) with the corresponding element in matrix \(C\). The result was another matrix \(\left[\begin{array}{rrr}7 & 6 & 4 \ 11 & -3 & -6 \ 11 & 7 & -2 \end{array}\right]\).
It's important to ensure both matrices are of the same size, otherwise, addition cannot be performed.

Matrix Multiplication

Matrix multiplication is slightly more complex than matrix addition. This involves taking the dot product of rows and columns. When multiplying two matrices \(A\) and \(B\), we calculate each element of the resulting matrix by:

• Taking a row from matrix \(A\) and a column from matrix \(B\), multiplying corresponding elements, and summing them up.
• Repeat this process for each row in \(A\) and each column in \(B\).

For a matrix product to be well-defined, the number of columns in the first matrix must equal the number of rows in the second matrix. In the exercise, we performed matrix multiplications like \(A(B + C)\) and found that they both required systematic computations to match the dimensions. Ensuring the dimensionality aligns is crucial for valid matrix multiplication.

Distributive Property

The distributive property is a fundamental algebraic rule that applies to matrix operations, akin to variables in arithmetic. This property in matrices means that multiplying a matrix by a sum of matrices can be distributed:

• \(A(B + C) = AB + AC\)

In essence, you distribute matrix \(A\) across the sum \(B + C\). In the exercise, this property was tested by comparing \(A(B + C)\) and \(AB + AC\). Both results should be identical if calculations are performed accurately, demonstrating the validity of the property. This property is crucial as it simplifies the calculation process in larger computations.

Graphing Calculator

A graphing calculator is a powerful tool for performing complex mathematical operations, such as matrix calculations. These devices can help visualize and compute tasks like matrix addition and multiplication that might be cumbersome by hand:

• Allows input of matrices and calculates results efficiently.
• Useful for verifying computed results in exercises, speeding up calculations.

While learning matrix operations, using a graphing calculator can reinforce understanding by allowing fast experimentation and correction. In the exercise, graphing calculators could verify results after manually performing operations, ensuring both understanding and accuracy. They bridge the gap between theoretical comprehension and practical application.