Matrices

Lesson 1: Basic Matrix Operations

• Adding and Subtracting Matrices
• A matrix is a rectangular arrangement of numbers
• The dimensions of a matrix with m rows and n columns are m × n (read "m by n")
• The numbers in a matrix are its elements
• Two matrices are equal when their dimensions are the same and the elements are equal
• To add or subtract two matrices, add or subtract their corresponding elements
  • Add or subtract matrices only when they have the same dimensions
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• Scalar Multiplication
• When working with matrices, a real number is called a scalar
• To multiply a matrix by a scalar, multiply each element in the matrix by the scalar
• This is called scalar multiplication
• Scalar Matrix Equations
• Using the definition of equal matrices, you can equate the elements in corresponding portions of two matrices that are equal

Lesson 2: Multiplying Matrices

• Multiplying Matrices
• The product of two matrices A  and B is defined provided the number of columns in A is equal to the number of rows in B
• If A is an m × n matrix and B is an n × p matrix, then the product AB is an m × p matrix
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• To find the element in the ith row and jth column of the product matrix AB, multiply each element in the ith row of A by the corresponding element in the jth column of B, then add the products
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• Matrix multiplication is not commutative
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Lesson 3: Matrix Determinants and Cramer's Rule

• The Determinant of a Matrix
• Determinant of a 2x2 Matrix
  • The determinant of a 2x2 matrix is the difference of the products of the elements on the diagonals shown
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• Determinant of a 3x3 Matrix
  • Repeat the first two columns to the right of the determinant
  • Subtract the sum of the red products from the sum of the blue products
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• Cramer's Rule
  • You can use determinants to solve a system of linear equations
  • The method, called Cramer's Rule, uses the coefficient matrix of the linear system
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  • Cramer's Rule for a 2x2 System
  • Let A be the coefficient matrix of the linear system
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  • If det A ≠ 0, then the system has exactly one solution:
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  • Note that the numerators for x and y are the determinants of the matrices formed by replacing the values of the x and y coefficient columns, respectively, with the columns of the constant values
  • Cramer's Rule for a 3x3 System
  • Let A be the coefficient matrix of the linear system shown below
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  • If det A ≠ 0, then the system has exactly one solution:
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Lesson 4: Inverse Matrices

• Finding Inverse Matrices

• The n x n identity matrix is a matrix with ones for all elements on the main diagonal (top left to bottom right) and zeros for all the other elements

• If A is any n x n matrix and I is the n x n identity matrix, then AI = A and IA = A

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• Two n x n matrices A and B are inverses of each other when their product is the n x n identity matrix; AB = I and BA = I

• An n x n matrix A has an inverse if and only if det A ≠ 0

• The inverse of A is denoted by A-1

• The Inverse of a 2x2 Matrix

  

• Using an Inverse Matrix to Solve a Linear System

• Write the system as a matrix equation AX = B; the matrix A is the coefficient matrix, X is the matrix of variables, and B is the matrix of constants

• Find the inverse of matrix A

• Multiply each side of AX = B by A-1 on the left to find the solution X = A-1B