Q: Given matrices $$A = \begin{pmatrix} 3 & 4 & 5 \\ 1 & 2 & 3 \end{pmatrix}$$ and $$B = \begin{pmatrix} 1 & -2 & 4 \\ -2 & -4 & 8 \end{pmatrix}$$, what is $$A + B$$?

$$\begin{pmatrix} 4 & 2 & 9 \\ -1 & -2 & 11 \end{pmatrix}$$

Q: If $$A = \begin{pmatrix} 3 & 4 & 5 \\ 1 & 2 & 3 \end{pmatrix}$$ and $$B = \begin{pmatrix} 1 & -2 & 4 \\ -2 & -4 & 8 \end{pmatrix}$$, what is $$A - B$$?

$$\begin{pmatrix} 2 & 6 & 1 \\ 3 & 6 & -5 \end{pmatrix}$$

Q: A business tracks monthly sales for two stores and three products. If January sales are $$\begin{pmatrix} 50 & 180 \\ 0 & 60 \end{pmatrix}$$, February sales are $$\begin{pmatrix} 40 & 0 \\ 100 & 120 \end{pmatrix}$$, and March sales are $$\begin{pmatrix} 0 & 50 \\ 0 & 150 \end{pmatrix}$$, what is the total sales for the first quarter?

$$\begin{pmatrix} 90 & 230 \\ 100 & 330 \end{pmatrix}$$

Q: Given $$A = \begin{pmatrix} 3 & 4 \\ 1 & 2 \end{pmatrix}$$ and $$B = \begin{pmatrix} 1 & -2 \\ -2 & -4 \end{pmatrix}$$, what is $$AB$$?

$$\begin{pmatrix} -5 & -22 \\ -3 & -10 \end{pmatrix}$$

Q: In a real-world scenario, a company wants to calculate total production costs using the following matrices: labor and material quantities for each plant and year, and unit costs for labor and materials. Which formula correctly represents the calculation?

$$\text{Total Cost} = \text{Quantity Matrix} \times \text{Unit Cost Matrix}$$

Question 1

Which of the following statements about matrix addition is correct?

Accepted Answer

Matrices can only be added if they have the same dimensions.

Question 2

Given matrices $$A = \begin{pmatrix} 3 & 4 & 5 \ 1 & 2 & 3 \end{pmatrix}$$ and $$B = \begin{pmatrix} 1 & -2 & 4 \ -2 & -4 & 8 \end{pmatrix}$$, what is $$A + B$$?

Accepted Answer

$$\begin{pmatrix} 4 & 2 & 9 \ -1 & -2 & 11 \end{pmatrix}$$

Question 3

If $$A = \begin{pmatrix} 3 & 4 & 5 \ 1 & 2 & 3 \end{pmatrix}$$ and $$B = \begin{pmatrix} 1 & -2 & 4 \ -2 & -4 & 8 \end{pmatrix}$$, what is $$A - B$$?

Accepted Answer

$$\begin{pmatrix} 2 & 6 & 1 \ 3 & 6 & -5 \end{pmatrix}$$

Question 4

A business tracks monthly sales for two stores and three products. If January sales are $$\begin{pmatrix} 50 & 180 \ 0 & 60 \end{pmatrix}$$, February sales are $$\begin{pmatrix} 40 & 0 \ 100 & 120 \end{pmatrix}$$, and March sales are $$\begin{pmatrix} 0 & 50 \ 0 & 150 \end{pmatrix}$$, what is the total sales for the first quarter?

Accepted Answer

$$\begin{pmatrix} 90 & 230 \ 100 & 330 \end{pmatrix}$$

Question 5

Which operation is required to find the product costs at each plant for each year, given a matrix of labor and material quantities and a matrix of unit costs?

Accepted Answer

Matrix multiplication

Question 6

Given $$A = \begin{pmatrix} 3 & 4 \ 1 & 2 \end{pmatrix}$$ and $$B = \begin{pmatrix} 1 & -2 \ -2 & -4 \end{pmatrix}$$, what is $$AB$$?

Accepted Answer

$$\begin{pmatrix} -5 & -22 \ -3 & -10 \end{pmatrix}$$

Question 7

If a matrix $$A is$$ $$2 	imes 3$$ and matrix $$B is$$ $$2 	imes 3$$, which of the following operations is NOT possible?

Accepted Answer

Multiplying $$A 	imes B$$

Question 8

In a real-world scenario, a company wants to calculate total production costs using the following matrices: labor and material quantities for each plant and year, and unit costs for labor and materials. Which formula correctly represents the calculation?

Accepted Answer

$$	ext{Total Cost} = 	ext{Quantity Matrix} 	imes 	ext{Unit Cost Matrix}$$

Question 9

Given $$X = \begin{pmatrix} 1 & 0.6 \ 1 & 0.8 \ 1 & 1 \ 1 & 1.2 \ 1 & 1.4 \ 1 & 1.6 \end{pmatrix}$$ and $$\vec{y} = \begin{pmatrix} 1.6 \ 1.2 \ 1.0 \ 0.6 \ 0.3 \ 0.4 \end{pmatrix}$$, which expression represents the least squares solution for $$\vec{\beta} in $$linear regression?

Accepted Answer

$$\vec{\beta} = (X^T X$$)^{-1}$$ X^T \vec{y}$$

Question 10

If $$M = I - X(X^T X$$)^{-1}$$ X^T$$, what does $$M$$ represent in the context of linear regression?

Accepted Answer

The projection matrix onto the residual space

Question 11

Which of the following is the correct formula for the projection matrix $$P$$ onto the column space of $$X$$?

Accepted Answer

$$P = X(X^T X$$)^{-1}$$ X^T$$

Question 12

If $$A is a$$ $$2 	imes 3$$ matrix and $$B is a$$ $$2 	imes 3$$ matrix, which operation is valid?

Accepted Answer

Adding $$A + B$$

Question 13

Suppose you have a matrix $$X$$ and vector $$\vec{y} as in $$the challenge question. What does $$\vec{y}^T \vec{y}$$ compute?

Accepted Answer

The sum of squares of the entries in $$\vec{y}$$

Question 14

Which of the following best describes the result of multiplying a $$2 	imes 2$$ matrix by another $$2 	imes 2$$ matrix?

Accepted Answer

A $$2 	imes 2$$ matrix

Question 15

A company wants to calculate the total wage and material costs for three plants in two years. If the wage and material rates are $$\begin{pmatrix} 25 & 20 \ 80 & 60 \end{pmatrix}$$ and the quantities for each plant are given, which operation should be used?

Accepted Answer

Multiply the quantity matrix by the rate matrix

Question 16

If $$A is a$$ $$2 	imes 2$$ matrix and $$B is a$$ $$2 	imes 2$$ matrix, which formula correctly computes $$AB$$?

Accepted Answer

$$AB = \begin{pmatrix} a$$_{11}$$b_{11} + a$$_{12}$$b_{21} & a$$_{11}$$b_{12} + a$$_{12}$$b_{22} \ a$$_{21}$$b_{11} + a$$_{22}$$b_{21} & a$$_{21}$$b_{12} + a$$_{22}$$b_{22} \end{pmatrix}$$

Question 17

In the context of business calculus, why is matrix multiplication used to calculate production costs?

Accepted Answer

It allows combining quantities and unit costs to get total costs for each product and plant.

Question 18

If $$X is a $$matrix of predictors and $$\vec{y} is a $$vector of outcomes, what does $$X^T \vec{y}$$ represent in regression analysis?

Accepted Answer

The vector of covariances between predictors and outcomes

Question 19

Which of the following is NOT a valid operation for two matrices of different sizes?

Accepted Answer

Addition

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

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