Step-by-Step Guidance for Matrix Algebra in Business Calculus

Q1a. Determine if the sum of the following matrices is possible:

Matrix 1:

$\begin{bmatrix}3 & 4 & 5 \\ 6 & 1 & 2 \\ 3 & 4\end{bmatrix}$

Matrix 2:

$\begin{bmatrix}1 & -2 & 4 \\ -2 & -4 & 8\end{bmatrix}$

Background

Topic: Matrix Addition

This question tests your understanding of when two matrices can be added together, which is a foundational concept in matrix algebra used in business applications such as economics and management science.

Key Terms and Formulas:

• Matrix Addition: Two matrices can be added only if they have the same dimensions (i.e., the same number of rows and columns).

Step-by-Step Guidance

1. Check the dimensions of both matrices. Count the number of rows and columns in each matrix.

2. Compare the dimensions. If both matrices have the same number of rows and columns, addition is possible; otherwise, it is not.

Try solving on your own before revealing the answer!

Final Answer:

The matrices have different dimensions, so their sum does not exist.

Q1b. Find the sum of the following matrices:

Matrix 1:

$\begin{bmatrix}3 & 4 & 5 \\ 1 & 2 & 3\end{bmatrix}$

Matrix 2:

$\begin{bmatrix}1 & -2 & 4 \\ -2 & -4 & 8\end{bmatrix}$

Background

Topic: Matrix Addition

This question tests your ability to add two matrices of the same size by adding their corresponding elements.

Key Terms and Formulas:

• Matrix Addition: $A + B = [a_{ij} + b_{ij}]$ where $a_{ij}$ and $b_{ij}$ are the elements in the $i$th row and $j$th column of matrices $A$ and $B$ respectively.

Step-by-Step Guidance

1. Verify that both matrices have the same dimensions (2 rows, 3 columns).

2. Add the corresponding elements from each matrix to form a new matrix. For example, add the element in row 1, column 1 of Matrix 1 to the element in row 1, column 1 of Matrix 2.

3. Continue this process for all elements in the matrices.

4. Write out the resulting matrix, but do not compute the final values yet.

Try solving on your own before revealing the answer!

Final Answer:

$\begin{bmatrix}3+1 & 4+(-2) & 5+4 \\ 1+(-2) & 2+(-4) & 3+8\end{bmatrix} = \begin{bmatrix}4 & 2 & 9 \\ -1 & -2 & 11\end{bmatrix}$

Each element is the sum of the corresponding elements from the two matrices.

Q2. Calculate the difference of the following matrices:

Matrix 1:

$\begin{bmatrix}3 & 4 & 5 \\ 1 & 2 & 3\end{bmatrix}$

Matrix 2:

$\begin{bmatrix}1 & -2 & 4 \\ -2 & -4 & 8\end{bmatrix}$

Background

Topic: Matrix Subtraction

This question tests your ability to subtract one matrix from another by subtracting corresponding elements, which is useful in business for comparing data sets.

Key Terms and Formulas:

• Matrix Subtraction: $A - B = [a_{ij} - b_{ij}]$

Step-by-Step Guidance

1. Confirm that both matrices have the same dimensions.

2. Subtract each element in Matrix 2 from the corresponding element in Matrix 1.

3. Write out the resulting matrix with the subtraction expressions for each element.

Try solving on your own before revealing the answer!

Final Answer:

$\begin{bmatrix}3-1 & 4-(-2) & 5-4 \\ 1-(-2) & 2-(-4) & 3-8\end{bmatrix} = \begin{bmatrix}2 & 6 & 1 \\ 3 & 6 & -5\end{bmatrix}$

Each element is the difference of the corresponding elements from the two matrices.

Q3. Use matrix addition to find the first quarter sales for each product at each store from the following monthly amounts:

January:

$\begin{bmatrix}50 & 180 \\ 0 & 60\end{bmatrix}$

February:

$\begin{bmatrix}40 & 0 \\ 100 & 120\end{bmatrix}$

March:

$\begin{bmatrix}0 & 50 \\ 0 & 150\end{bmatrix}$

Background

Topic: Matrix Addition in Business Applications

This question tests your ability to sum matrices representing monthly sales data to find quarterly totals, a common business application.

Key Terms and Formulas:

• Matrix Addition: Add corresponding elements from each matrix to get the total for each product at each store.

Step-by-Step Guidance

1. Check that all three matrices have the same dimensions.

2. Add the corresponding elements from the January, February, and March matrices to get the total for each position.

3. Write out the resulting matrix with the sum expressions for each element.

Try solving on your own before revealing the answer!

Final Answer:

$\begin{bmatrix}50+40+0 & 180+0+50 \\ 0+100+0 & 60+120+150\end{bmatrix} = \begin{bmatrix}90 & 230 \\ 100 & 330\end{bmatrix}$

This matrix gives the first quarter sales for each product at each store.

Q4. Find the product costs at each plant for each year using matrix multiplication:

Matrix of workers and materials (rows: plants, columns: workers and materials):

$\begin{bmatrix}20 & 80 \\ 40 & 100 \\ 30 & 90\end{bmatrix}$

Matrix of costs (columns: years 2002 and 2003):

$\begin{bmatrix}25 & 20 \\ 80 & 60\end{bmatrix}$

Background

Topic: Matrix Multiplication in Business Applications

This question tests your ability to use matrix multiplication to calculate total costs, a key skill in business calculus for analyzing production costs.

Key Terms and Formulas:

• Matrix Multiplication: If $A$ is an $m \times n$ matrix and $B$ is an $n \times p$ matrix, their product $AB$ is an $m \times p$ matrix where each entry is the sum of products of corresponding entries from the row of $A$ and the column of $B$.

Step-by-Step Guidance

1. Check that the number of columns in the first matrix matches the number of rows in the second matrix (both are 2).

2. For each plant (row), multiply the number of workers by the wage rate and the amount of materials by the material cost for each year, then sum these products for each year.

3. Write out the multiplication expressions for each plant and year, but do not compute the final values yet.

Try solving on your own before revealing the answer!

Final Answer:

For each plant and year:

• Plant 1, 2002: $(20 \times 25) + (80 \times 80)$

• Plant 1, 2003: $(20 \times 20) + (80 \times 60)$

• Plant 2, 2002: $(40 \times 25) + (100 \times 80)$

• Plant 2, 2003: $(40 \times 20) + (100 \times 60)$

• Plant 3, 2002: $(30 \times 25) + (90 \times 80)$

• Plant 3, 2003: $(30 \times 20) + (90 \times 60)$

Final matrix:

$\begin{bmatrix}6900 & 5200 \\ 9000 & 6800 \\ 7950 & 6000\end{bmatrix}$

Each entry represents the total cost for a plant in a given year.

Q5. Calculate the product $AB$ for the following matrices:

$A = \begin{bmatrix}3 & 4 \\ 1 & 2\end{bmatrix}$, $B = \begin{bmatrix}1 & -2 \\ -2 & -4\end{bmatrix}$

Background

Topic: Matrix Multiplication

This question tests your ability to multiply two $2 \times 2$ matrices, a fundamental operation in business calculus for modeling systems and transformations.

Key Terms and Formulas:

• Matrix Multiplication for $2 \times 2$ matrices:

  If $A = \begin{bmatrix}a & b \\ c & d\end{bmatrix}$ and $B = \begin{bmatrix}e & f \\ g & h\end{bmatrix}$, then $AB = \begin{bmatrix}ae + bg & af + bh \\ ce + dg & cf + dh\end{bmatrix}$

Step-by-Step Guidance

1. Write out the formula for each entry in the product matrix $AB$.

2. For the entry in row 1, column 1: Multiply the first row of $A$ by the first column of $B$ and sum the products.

3. For the entry in row 1, column 2: Multiply the first row of $A$ by the second column of $B$ and sum the products.

4. Repeat for row 2 of $A$ with both columns of $B$.

5. Write out the resulting matrix with the multiplication and addition expressions, but do not compute the final values yet.

Try solving on your own before revealing the answer!

Final Answer:

$AB = \begin{bmatrix}(3 \times 1) + (4 \times -2) & (3 \times -2) + (4 \times -4) \\ (1 \times 1) + (2 \times -2) & (1 \times -2) + (2 \times -4)\end{bmatrix} = \begin{bmatrix}-5 & -22 \\ -3 & -10\end{bmatrix}$

Each entry is calculated by multiplying and summing the appropriate elements from $A$ and $B$.

Challenge Question: Perform the indicated computations with the following matrices:

$X = \begin{bmatrix}1 & 0.6 \\ 1 & 0.8 \\ 1 & 1.0 \\ 1 & 1.2 \\ 1 & 1.4 \\ 1 & 1.6\end{bmatrix}$, $\vec{y} = \begin{bmatrix}1.6 \\ 1.2 \\ 1.0 \\ 0.6 \\ 0.3 \\ 0.4\end{bmatrix}$

1. $X^T X$

2. $(X^T X)^{-1}$

3. $X^T \vec{y}$

4. $(X^T X)^{-1} X^T \vec{y}$

5. $\vec{y}^T \vec{y}$

6. $M = I - X(X^T X)^{-1} X^T$

7. $P = I - M$

Background

Topic: Matrix Algebra in Linear Regression

This challenge question tests your ability to perform advanced matrix operations used in the least squares method for linear regression, a key application in business analytics.

Key Terms and Formulas:

• Transpose: $X^T$ is the transpose of $X$.

• Matrix Multiplication: $X^T X$ is the product of $X^T$ and $X$.

• Matrix Inverse: $(X^T X)^{-1}$ is the inverse of $X^T X$.

• Projection Matrix: $M = I - X(X^T X)^{-1} X^T$

Step-by-Step Guidance

1. Compute $X^T$, the transpose of $X$.

2. Multiply $X^T$ by $X$ to get $X^T X$.

3. Find the inverse of $X^T X$, denoted $(X^T X)^{-1}$.

4. Multiply $X^T$ by $\vec{y}$ to get $X^T \vec{y}$.

5. Multiply $(X^T X)^{-1}$ by $X^T \vec{y}$ to get the least squares solution vector.

6. Compute $\vec{y}^T \vec{y}$ (the sum of squares of $\vec{y}$).

7. Construct the identity matrix $I$ of appropriate size, then compute $M = I - X(X^T X)^{-1} X^T$.

8. Finally, compute $P = I - M$.

Try solving on your own before revealing the answer!

Final Answer:

See the provided Challenge Question Solution.pdf for detailed calculations and final results for each part (a) through (g). The steps above outline the process for each computation.