Determinants Formulas

Posted In: Mathematics

In linear algebra, the determinant is a value that can be computed from the elements of a square matrix. The determinant of a matrix A is denoted det(A), det A, or |A|.

Matrices: A
Elements of a matrix: ai, ai, aij, bij, … aij
Determinant of matrix A: |A|

  1. Second order determinant
    $|A|=|\table a_1,b_1;a_2,b_2|=a_1b_2-a_2b_1$
  2. Third order determinant
    $|A|=|\table a_11,a_12,a_13;a_21,a_22,a_23;a_31,a_32,a_33|$
    $=a_11a_22a_33+a_12a_23a_31+a_13a_21a_32-a_11a_23a_32-a_12a_21a_33-a_13a_22a_31$
    Note: Above equation can be remembered easily by Sarrus Rule (Arrow Rule)
  3. Nth order determinants
    $|A|=|\table a_11, a_12, …, a_1j, …, a_1n;b_11, b_12, …, b_1j, …, b_1n;…, …, …, …, …, …;…, …, …, …, …, …;a_{i1}, a_{i2}, …, a_{ij}, …, a_{in};…, …, …, …, …, …;a_{n1}, a_{n2}, …, a_{nj}, …, a_{nn}|$
  4. Minor
    The minor $M_{ij}$ associated with the element $a_{ij}$ of the $n-th$ order matrix $A$ is the (n-1)-th order determinant derived from the matrix $A$ by deletion of its i-th row and j-th column.
  5. Cofactor
    $C_{ij}=(-1)^{i+j}M_{ij}$
  6. Laplace Expansion of n-th Order Determinant
    Laplace expansion by elements of the i-th row
    $|A|=∑↙{j=1}↖na_{ij}\ C_{ij}, i=1,2,…,n$
    Laplace expansion by elements of the j-th column
    $|A|=∑↙{i=1}↖na_{ij}\ C_{ij}, j=1,2,…,n$



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