Matrices and Determinants: Properties and Operations
1. Determinant of a Matrix: The determinant of a matrix is equal to the determinant of its transpose.
2. Zero Row or Column: If a square matrix has a row or column of zeros, its determinant is 0.
3. Swapping Parallel Lines: Swapping two rows or columns of a square matrix changes the sign of its determinant.
4. Identical Parallel Lines: If a square matrix has two identical rows or columns, its determinant is 0.
5. Multiplying a Row/Column: Multiplying all elements of a row or column of a square matrix by a scalar multiplies the determinant by that scalar.
6. Proportional Rows or Columns: If a matrix has two proportional rows or columns, its determinant is 0.
7. Determinant Decomposition: The determinant of a sum of two matrices can be decomposed into the sum of the determinants of the individual matrices. For example, in a 2×2 matrix: det(A + B) = det(A) + det(B). In a 3×3 matrix: det(A + B) = det(A) + det(B) + other terms.
8. Linear Combination of Rows: Adding a linear combination of other rows to a row of a matrix does not change the determinant.
9. Linearly Dependent Row: If a row of a matrix is a linear combination of other rows, the determinant is 0.
10. Product of Matrices: The determinant of the product of two matrices equals the product of their determinants: det(AB) = det(A) * det(B).
11. Multiplying by Adjoints (3×3): Multiplying the elements of a row or column of a 3×3 matrix by their adjoints and summing the results yields the determinant of the original matrix.
12. Multiplying by Elements of a Parallel Line (3×3): Multiplying the elements of a row or column by the corresponding elements of a parallel row or column and summing the results yields 0.
Note: det(A) + det(B) is not generally equal to det(A + B).
External Properties of Matrices:
- Associative Property: (AB)C = A(BC)
- Distributive Property: (A + B)C = AC + BC and A(C + B) = AC + AB
- Identity Matrix: AI = IA = A
- Inverse Matrix: AA-1 = A-1A = I
- Scalar Multiplication: det(kA) = kn * det(A), where n is the order of the matrix.
Determinant of a Square Matrix of Order ‘n’: The determinant of a square matrix of order ‘n’ is a sum of n! terms. Each term is a product of n elements, one from each row and each column. The sign of each term depends on the number of inversions in the column indices when the row indices are in natural order. An even number of inversions results in a positive sign, and an odd number results in a negative sign.
Minor and Cofactor:
- Minor (Mij): The determinant of the (n-1) x (n-1) matrix obtained by deleting the i-th row and j-th column of the original matrix.
- Cofactor (Cij): Cij = (-1)i+j * Mij
Adjoint (Adj(A)): The transpose of the matrix of cofactors.
Rank of a Matrix: The number of linearly independent rows (or columns) of a matrix. A matrix with determinant 0 has linearly dependent rows or columns.
Minor of Order h of an m x n Matrix: The determinant of an h x h submatrix formed by selecting h rows and h columns.
Invertible Matrix: A square matrix with a non-zero determinant. An invertible matrix has an inverse.
Linear Equation and System of Linear Equations:
- Linear Equation: A polynomial equation of degree 1 in one or more unknowns.
- System of Linear Equations: A set of linear equations.
- Homogeneous System: A system of linear equations where all constant terms are 0.
Matrix Types:
- Square Matrix: Number of rows equals the number of columns.
- Rectangular Matrix: Number of rows is not equal to the number of columns.
- Row Matrix: A matrix with only one row.
- Column Matrix: A matrix with only one column.
- Transpose Matrix: Rows and columns are interchanged.
- Null Matrix: All elements are 0.
- Diagonal Matrix: All elements outside the main diagonal are 0.
- Triangular Matrix: Elements above or below the main diagonal are 0.
- Identity Matrix: A diagonal matrix with all diagonal elements equal to 1.
- Symmetric Matrix: A square matrix where Aij = Aji.
- Antisymmetric Matrix: A square matrix where Aij = -Aji and diagonal elements are 0.
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