Matrices and Determinants Study Cards

  • Matrices Definition: A set of m x n numbers arranged in a rectangular array with m rows and n columns.

  • Reading Order: Read as "m by n."

  • General Representation: A = \begin{pmatrix} a{11} & a{12} & \dots & a{1n} \ a{21} & a{22} & \dots & a{2n} \
    \vdots & \vdots & \ddots & \vdots \
    a{m1} & a{m2} & \dots & a_{mn} \end{pmatrix}

  • Notation: Denoted as A = [a{ij}]{m x n}.

  • Types of Matrices:

    • Row Matrix: 1 x n

    • Column Matrix: m x 1

    • Zero Matrix: All elements zero

    • Singleton Matrix: 1 x 1

    • Rectangular Matrix: m ≠ n

    • Square Matrix: m = n

    • Diagonal Matrix: Non-diagonal elements zero

    • Identity Matrix: Diagonal elements are 1

  • Standard Values:

    • Zero Matrix (O): All elements are zero. For example, O for a 2x2 matrix is \begin{pmatrix} 0 & 0 \
      0 & 0 \end{pmatrix}.

    • Identity Matrix (I): For 2x2, I = \begin{pmatrix} 1 & 0 \
      0 & 1 \end{pmatrix}. For 3x3, I = \begin{pmatrix} 1 & 0 & 0 \
      0 & 1 & 0 \
      0 & 0 & 1 \end{pmatrix}.

  • Matrix Properties:

    • Addition: Only for matrices of the same order; corresponding elements are added.

    • Scalar Multiplication: A scalar k is multiplied by every element of the matrix.

    • Matrix Multiplication:

    • A (order m x n) can multiply B (order p x q) if n = p.

    • AB ≠ BA (Generally non-commutative).

    • (AB)C = A(BC) (Associative Law).

    • A(B + C) = AB + AC (Distributive Law).

    • Equality of Matrices: Matrices are equal if they have the same order, and all corresponding elements are equal.

  • Determinants:

    • A number associated with every square matrix, denoted as |A| or det(A).

    • Minor (M_{ij}): Determinant obtained by deleting the i-th row and j-th column.

    • Cofactor (C{ij}): C{ij} = (-1)^{i+j} M_{ij}.

    • Properties of Determinants:

    • |A| = |A^T|.

    • Interchanging two rows/columns changes the sign of |A|.

    • If two rows/columns are identical, |A| = 0.

    • If all elements of a row/column are zero, |A| = 0.

    • |kA| = k^n |A| where n is the order.

    • |AB| = |A| |B|.

  • Characteristic Equation:

    • |A - λ I| = 0; the roots λ are characteristic roots (eigenvalues).

    • Cayley-Hamilton Theorem: Every square matrix satisfies its own characteristic equation.