4.5 Modeling change in a context using matrices

• Rows in a matrix are arranged horizontally
• Match the property of matrix multiplication with its description:

  Associativity ↔️ (AB)C = A(BC)

  Distributivity ↔️ A(B + C) = AB + AC

  Identity matrix ↔️ AI = A
• Steps to perform matrix multiplication:

  1️⃣ Ensure the number of columns in the first matrix matches the number of rows in the second matrix

  2️⃣ Calculate each element (i, j) in the resulting matrix

  3️⃣ Sum the products of corresponding elements
• Matrices can be used to represent and analyze linear transformations.

  True
• What are matrices used to analyze in 2D or 3D space?

  Linear transformations
• Match the context with its matrix application:

  Economy ↔️ Input-output analysis

  Digital images ↔️ Image processing

  Social networks ↔️ Network analysis
• Steps to construct a matrix to represent changes:

  1️⃣ Identify the variables

  2️⃣ Arrange the variables as rows and columns

  3️⃣ Determine the rates of change

  4️⃣ Create a matrix with these rates
• A population migration matrix shows the movement of people between cities
• Rows in a matrix are arranged horizontally.


  True
• Steps to perform matrix multiplication:

  1️⃣ Ensure the number of columns in the first matrix equals the number of rows in the second matrix

  2️⃣ Calculate each element (i, j) in the resulting matrix
• What condition must be met to multiply two matrices?

  Columns of A = rows of B
• In which context are matrices used to represent rotations and scaling in 2D or 3D space?

  Linear transformations
• An adjacency matrix is used in network analysis to model connections in networks.

  True
• To construct a matrix for changes, the first step is to identify the variables
• When modeling changes over time using matrices, the rates of change represent how variables are changing relative to one another
• Steps to apply matrix multiplication for modeling changes over time

  1️⃣ Identify the variables that are changing

  2️⃣ Arrange the variables as rows and columns of a matrix

  3️⃣ Determine the rates of change for each variable

  4️⃣ Construct a matrix with these rates of change as elements
• Non-commutativity in matrix multiplication means the order of multiplication does not matter.

  False
• Examining the transition matrix helps understand the rates of change between variables.


  True
• The associative property allows modeling multiple transitions in sequence.


  True
• What is a matrix defined as?

  A rectangular array of numbers
• What is matrix multiplication calculated by summing?

  Products of corresponding elements
• Matrix multiplication is commutative.

  False
• Key components of a matrix include rows, columns, and elements.


  True
• What is one application of matrices in linear transformations?

  Analyzing rotations in 2D space
• Match the matrix application with its description:

  Markov Chains ↔️ Model probabilities of state transitions

  Input-Output Analysis ↔️ Model interdependencies between sectors

  Network Analysis ↔️ Model relationships in networks
• Matrices can model the probabilities of transitioning between states in a Markov chain.

  True
• To construct matrices to represent changes, you must first identify the variables
• What type of matrix shows the movement of people between cities?

  Population migration matrix
• What is a matrix fundamentally used for in linear algebra?

  Represent linear transformations
• What is the result of multiplying two matrices called?

  New matrix
• Matrix multiplication is commutative.

  False
• AI = IA = A, where I is the identity matrix.


  True
• A transition matrix in a Markov chain models the probabilities of transitioning between states
• What is the third step in constructing matrices to represent changes?

  Determine rates of change
• What type of variables can be modeled using matrix multiplication to show changes over time?

  Population, inventory, returns
• Matrix multiplication is a powerful tool for modeling changes over time
• The associative property of matrix multiplication allows chaining multiple transitions
• What is the next step after constructing a matrix model for changes over time?

  Analyze and interpret the model
• Multiplying the transition matrix by the initial state projects how the variables will change over time
• Elements in a matrix are individual numbers within the matrix.


  True