Matrix Operations Quiz

Matrices

Matrix Operations

• Addition:

  • Matrices of the same dimension can be added.
  • Element-wise addition: ( C[i][j] = A[i][j] + B[i][j] ).

• Subtraction:

  • Similar to addition, matrices of the same dimension can be subtracted.
  • Element-wise subtraction: ( C[i][j] = A[i][j] - B[i][j] ).

• Scalar Multiplication:

  • Each element of the matrix is multiplied by a scalar ( k ).
  • ( B[i][j] = k \cdot A[i][j] ).

• Matrix Multiplication:

  • Requires that the number of columns in the first matrix matches the number of rows in the second.
  • Resulting matrix ( C ) with dimensions ( m \times p ) where ( A ) is ( m \times n ) and ( B ) is ( n \times p ).
  • Calculated as: ( C[i][j] = \sum_{k=1}^{n} A[i][k] \cdot B[k][j] ).

• Transpose of a Matrix:

  • Flips a matrix over its diagonal.
  • ( (A^T)[i][j] = A[j][i] ).

• Inverse of a Matrix:

  • A matrix ( A ) has an inverse ( A^{-1} ) if ( A \cdot A^{-1} = I ) (identity matrix).
  • Only square matrices can have inverses.
  • Inverse can be calculated using methods such as Gauss-Jordan elimination or using determinants.

Determinants

• Definition:

  • A scalar value derived from a square matrix that provides information about the matrix (e.g., whether it is invertible).

• Properties:

  • If the determinant is zero, the matrix is singular (non-invertible).
  • The determinant changes sign if two rows (or columns) are swapped.
  • Scaling a row by a factor ( k ) scales the determinant by ( k ).
  • The determinant of an identity matrix is 1.

• Calculation:

  • 2x2 Matrix: For ( A = \begin{pmatrix} a & b \ c & d \end{pmatrix} ), ( \text{det}(A) = ad - bc ).
  • 3x3 Matrix: For ( A = \begin{pmatrix} a & b & c \ d & e & f \ g & h & i \end{pmatrix} ),
    • ( \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) ).
  • Larger Matrices: Use methods such as:
    • Expansion by minors and cofactors.
    • Row reduction to convert to upper triangular form and then multiply the diagonal entries.

• Applications:

  • Used in solving systems of linear equations (Cramer's rule).
  • Helps in finding eigenvalues and eigenvectors.

Matrix Operations

• Matrices can be added or subtracted only if they have the same dimensions through element-wise operations.
• Scalar multiplication involves multiplying each element of a matrix by a scalar ( k ), resulting in a new matrix.
• Matrix multiplication requires the number of columns in the first matrix to equal the number of rows in the second, yielding a product matrix ( C ) with dimensions ( m \times p ).
• The entry in the resulting matrix ( C[i][j] ) is calculated by summing products of corresponding elements from the rows of ( A ) and columns of ( B ).
• The transpose, denoted ( A^T ), of a matrix flips it over its diagonal, switching rows and columns: ( (A^T)[i][j] = A[j][i] ).
• A matrix ( A ) possesses an inverse ( A^{-1} ) if their product yields the identity matrix ( I ); only square matrices can have inverses.
• Methods to calculate the inverse include Gauss-Jordan elimination and determinant evaluation.

Determinants

• The determinant is a scalar value associated with a square matrix, indicating properties like invertibility.
• A determinant of zero signals that the matrix is singular and non-invertible.
• The sign of the determinant reverses when two rows or columns are interchanged.
• Scaling a row by a constant ( k ) results in the determinant being scaled by the same factor.
• The determinant of an identity matrix is always 1.
• For a 2x2 matrix ( A = \begin{pmatrix} a & b \ c & d \end{pmatrix} ), the determinant is calculated as ( \text{det}(A) = ad - bc ).
• For a 3x3 matrix ( A = \begin{pmatrix} a & b & c \ d & e & f \ g & h & i \end{pmatrix} ), the determinant is ( \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) ).
• Determinants of larger matrices can be computed using techniques like expansion by minors or row reduction to upper triangular form, followed by multiplying the diagonal entries.
• Determinants are crucial for solving systems of linear equations, particularly via Cramer’s rule, and are essential in finding eigenvalues and eigenvectors.

Matrix Inversion

• Inverse Matrix: For a matrix ( A ), its inverse ( A^{-1} ) satisfies ( A A^{-1} = I ), where ( I ) is the identity matrix.
• Existence Criteria: A matrix possesses an inverse only if it is square and has a non-zero determinant (( \det(A) \neq 0 )).
• Adjugate Method: The formula for the inverse is ( A^{-1} = \frac{1}{\det(A)} \text{adj}(A) ).
• Gauss-Jordan Elimination: To find the inverse, augment the matrix ( A ) with the identity matrix ( I ) and transform it to ( [I | A^{-1}] ).
• Cofactor Expansion: Calculate the determinant first, then use it alongside the adjugate to find the inverse.
• Inverse Properties:
  • The inverse of the inverse returns the original matrix: ( (A^{-1})^{-1} = A ).
  • The inverse of a product of matrices reverses the order: ( (AB)^{-1} = B^{-1}A^{-1} ).
  • The inverse of the transpose equals the transpose of the inverse: ( (A^T)^{-1} = (A^{-1})^T ).

Matrix Operations

• Addition: Only valid for matrices with identical dimensions; resultant matrix ( C ) has elements ( c_{ij} = a_{ij} + b_{ij} ).
• Subtraction: Similar rules as addition; resultant matrix ( C ) has elements ( c_{ij} = a_{ij} - b_{ij} ).
• Scalar Multiplication: Multiply every element of matrix ( A ) by a scalar ( k ): ( B = kA ), where each element is ( b_{ij} = k \cdot a_{ij} ).
• Matrix Multiplication: Non-commutative nature of matrices; ( C = AB ) is computed as ( c_{ij} = \sum_{k} a_{ik} b_{kj} ), requiring ( A )'s columns to match ( B )'s rows.
• Transpose: For matrix ( A ), the transpose ( A^T ) is defined as ( (A^T){ij} = a{ji} ).
• Transpose Properties:
  • Addition and scalar multiplication of transposed matrices follow: ( (A + B)^T = A^T + B^T ) and ( (kA)^T = kA^T ).
  • The transpose of a product reverses the order: ( (AB)^T = B^T A^T ).

Eigenvalues and Eigenvectors

• Eigenvalue: Denoted by ( \lambda ), it satisfies the equation ( Av = \lambda v ) for a non-zero vector ( v ).
• Eigenvector: A non-zero vector that, when multiplied by matrix ( A ), changes only by the scalar eigenvalue.
• Characteristic Equation: To find eigenvalues, solve ( \det(A - \lambda I) = 0 ).
• Calculation of Eigenvectors: For each eigenvalue ( \lambda ), substitute into ( (A - \lambda I)v = 0 ) to determine associated eigenvectors.
• Eigenvalue Properties:
  • The sum of a matrix's eigenvalues equals its trace.
  • The product of eigenvalues is equivalent to the matrix's determinant.
  • Eigenvectors corresponding to different eigenvalues are linearly independent.