Matrix Operations

Matrix Operations

• Addition: Two matrices can be added if they have the same dimensions. The corresponding elements are added.
• Scalar Multiplication: A matrix can be multiplied by a scalar. Each element of the matrix is multiplied by the scalar.
• Matrix Multiplication: Two matrices can be multiplied if the number of columns in the first matrix is equal to the number of rows in the second matrix. The resulting matrix has the same number of rows as the first matrix and the same number of columns as the second matrix.

Determinant Properties

• Properties:
  • Linearity: The determinant is a linear function of each row (column) when the other rows (columns) are held constant.
  • Anti-commutativity: The determinant changes sign if two rows (columns) are interchanged.
  • Multiplicity: The determinant is multiplied by a scalar if a row (column) is multiplied by that scalar.
• Calculating Determinants:
  • 2x2 Matrix: det(A) = a(e - d) where A = [[a, b], [c, d]].
  • nxn Matrix: Can be calculated using the Laplace Expansion or Cofactor Expansion.

Inverse Matrices

• Definition: A matrix A has an inverse A^(-1) if A * A^(-1) = I, where I is the identity matrix.
• Properties:
  • (A * B)^(-1) = B^(-1) * A^(-1)
  • (A^(-1))^(-1) = A
  • A * A^(-1) = A^(-1) * A = I
• Calculating Inverse:
  • 2x2 Matrix: A^(-1) = (1 / det(A)) * [[d, -b], [-c, a]] where A = [[a, b], [c, d]].
  • nxn Matrix: Can be calculated using Gaussian Elimination or Cramer's Rule.

Eigenvalues

• Definition: A scalar λ is an eigenvalue of a matrix A if there exists a non-zero vector v such that A * v = λ * v.
• Properties:
  • Scalar Multiplication: A * v = λ * v implies A * (kv) = λ * (kv) for any scalar k.
  • Linear Independence: If v1 and v2 are eigenvectors corresponding to distinct eigenvalues, then they are linearly independent.
• Calculating Eigenvalues:
  • Characteristic Equation: det(A - λI) = 0 where A is the matrix and I is the identity matrix.
  • Eigenvalue Decomposition: A = Q * Λ * Q^(-1) where Q is an orthogonal matrix and Λ is a diagonal matrix of eigenvalues.

Matrix Operations

• Matrix addition is only possible when two matrices have the same dimensions.
• In matrix addition, corresponding elements are added.
• Scalar multiplication involves multiplying each element of a matrix by a scalar.
• Matrix multiplication is possible when the number of columns in the first matrix equals the number of rows in the second matrix.
• The resulting matrix from matrix multiplication has the same number of rows as the first matrix and the same number of columns as the second matrix.

Determinant Properties

• The determinant is a linear function of each row (column) when the other rows (columns) are held constant.
• The determinant changes sign when two rows (columns) are interchanged.
• The determinant is multiplied by a scalar when a row (column) is multiplied by that scalar.
• The determinant of a 2x2 matrix can be calculated using the formula det(A) = a(e - d) where A = [[a, b], [c, d]].
• The determinant of an nxn matrix can be calculated using the Laplace Expansion or Cofactor Expansion.

Inverse Matrices

• A matrix A has an inverse A^(-1) if A * A^(-1) = I, where I is the identity matrix.
• The inverse of a matrix product is the product of the inverses in reverse order: (A * B)^(-1) = B^(-1) * A^(-1).
• The inverse of an inverse matrix is the original matrix: (A^(-1))^(-1) = A.
• The product of a matrix and its inverse is the identity matrix: A * A^(-1) = A^(-1) * A = I.
• The inverse of a 2x2 matrix can be calculated using the formula A^(-1) = (1 / det(A)) * [[d, -b], [-c, a]] where A = [[a, b], [c, d]].
• The inverse of an nxn matrix can be calculated using Gaussian Elimination or Cramer's Rule.

Eigenvalues

• An eigenvalue is a scalar λ that satisfies the equation A * v = λ * v for a non-zero vector v.
• If v is an eigenvector of a matrix A with eigenvalue λ, then kv is also an eigenvector with eigenvalue λ for any scalar k.
• Eigenvectors corresponding to distinct eigenvalues are linearly independent.
• The characteristic equation det(A - λI) = 0 can be used to calculate eigenvalues.
• Eigenvalue decomposition is a representation of a matrix A as A = Q * Λ * Q^(-1) where Q is an orthogonal matrix and Λ is a diagonal matrix of eigenvalues.