Formulas and Tables

Matrices and Determinants

Properties of Determinants

Matrix: \(A\)
Elements of matrices: \({a_{ij}}\), \({a_i}\), \({b_i}\)
Determinant of a matrix: \(\det A\)
Minor of an element \({a_{ij}}\): \({M_{ij}}\)

Cofactor of an element \({a_{ij}}\): \({A_{ij}}\)
Real number: \(k\)
Natural numbers: \(n\), \(i\), \(j\), \(s\)

  1. The determinant of a square matrix \(\left[ {{a_{ij}}} \right]\) of order \(n\) is a polynomial composed of the elements of this matrix and containing \(n!\) terms of the form \({\left( { – 1} \right)^s}{a_{1{k_1}}}{a_{2{k_2}}} \cdots {a_{n{k_n}}}\). Each such term corresponds to one of \(n!\) different ordered \({k_1},{k_2}, \ldots {k_n}\) that result from \(s\) pairwise permutations of the elements of the set \(1,2, \ldots ,n\). The value of the determinant remains unchanged under linear combinations of rows or columns or transposition of the matrix.
  2. The determinant of an \(n\)th order matrix can be written as
    \(\det A =\) \( \left| {\begin{array}{*{20}{c}}
    {{a_{11}}}&{{a_{12}}}& \ldots &{{a_{1j}}}& \ldots &{{a_{1n}}}\\
    {{a_{21}}}&{{a_{22}}}& \ldots &{{a_{2j}}}& \ldots &{{a_{2n}}}\\
    \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\
    {{a_{i1}}}&{{a_{i2}}}& \ldots &{{a_{ij}}}& \ldots &{{a_{in}}}\\
    \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\
    {{a_{n1}}}&{{a_{n2}}}& \ldots &{{a_{nj}}}& \ldots &{{a_{nn}}}
    \end{array}} \right|\)
  3. Second order determinant
    The determinant of a second order matrix consists of \(2\) terms, each of which is the product of \(2\) elements:
    \(\det A = \left| {\begin{array}{*{20}{c}}
    {{a_{11}}}&{{a_{12}}}\\
    {{a_{21}}}&{{a_{22}}}
    \end{array}} \right| =\) \( {a_{11}}{a_{22}} – {a_{12}}{a_{21}}\)
  4. Third order determinant
    The determinant of a third order matrix includes \(6\) terms, each of which is the product of \(3\) elements:
    \(\det A =\) \( \left| {\begin{array}{*{20}{c}}
    {{a_{11}}}&{{a_{12}}}&{{a_{13}}}\\
    {{a_{21}}}&{{a_{22}}}&{{a_{23}}}\\
    {{a_{31}}}&{{a_{32}}}&{{a_{33}}}
    \end{array}} \right| =\) \( {a_{11}}{a_{22}}{a_{33}} \) \(+\; {a_{12}}{a_{23}}{a_{31}} \) \(+\; {a_{13}}{a_{21}}{a_{32}} \) \(-\; {a_{11}}{a_{23}}{a_{32}} \)
    \(-\; {a_{12}}{a_{21}}{a_{33}} \) \(-\; {a_{13}}{a_{22}}{a_{31}}\)
  5. The determinant of a third order matrix can also be calculated using the Sarrus’ rule. Three of the six terms are included with the plus sign, and the other three terms are taken with the minus sign. The corresponding triples of elements are shown schematically in the figure.
Sarrus’ rule (triples of elements with the plus sign)
Sarrus’ rule (triples of elements with the minus sign)
  1. Minor
    The first minor \({M_{ij}}\) associated with the element \({a_{ij}}\) of an \(n\)th order square matrix \(A\) is the determinant of order \(\left( {n – 1} \right)\) obtained from the matrix \(A\) by deleting the \(i\)th row and the \(j\)th column.
  2. Cofactor
    The cofactor \({A_{ij}}\) is the minor \({M_{ij}}\) multiplied by \(\left({-1}\right)\) raised to the \(\left({i + j}\right)\) power:
    \({A_{ij}} = {\left( { – 1} \right)^{i + j}}{M_{ij}}\)
  3. Laplace’s theorem
    An \(n\)th order determinant can be calculated using the Laplace’s formulas.
    Expansion of the determinant along the \(i\)th row is given by the formula
    \(\det A = \sum\limits_{j = 1}^n {{a_{ij}}{A_{ij}}} ,\;\) \(i = 1,2, \ldots ,n\)
    Expansion of the determinant along the \(j\)th column is expressed in the form
    \(\det A = \sum\limits_{i = 1}^n {{a_{ij}}{A_{ij}}} ,\;\;j = 1,2, \ldots ,n\)
  4. Determinant of a transpose
    The determinant of the transpose of a square matrix is equal to the determinant of the original matrix:
    \(\left| {\begin{array}{*{20}{c}}
    {{a_1}}&{{a_2}}\\
    {{b_1}}&{{b_2}}
    \end{array}} \right| =\) \( \left| {\begin{array}{*{20}{c}}
    {{a_1}}&{{b_1}}\\
    {{a_2}}&{{b_2}}
    \end{array}} \right|\)
  5. Reordering rows and columns in a determinant
    If two rows (or columns) are interchanged, the sign of the determinant is changed to the opposite:
    \(\left| {\begin{array}{*{20}{c}}
    {{a_1}}&{{b_1}}\\
    {{a_2}}&{{b_2}}
    \end{array}} \right| =\) \( -\left| {\begin{array}{*{20}{c}}
    {{a_2}}&{{b_2}}\\
    {{a_1}}&{{b_1}}
    \end{array}} \right|\)
  6. Determinant of a matrix with duplicate rows or columns
    If a matrix has two duplicate rows (or columns), the determinant of the matrix is zero:
    \(\left| {\begin{array}{*{20}{c}}
    {{a_1}}&{{a_1}}\\
    {{a_2}}&{{a_2}}
    \end{array}} \right| = 0\)
  7. Multiplication of a row (column) of a determinant by a constant
    Multiplication of the elements of any row (or column) by the same number is equivalent to multiplying the determinant by that number. In other words, a constant factor of the elements of any row (or column) can be taken out of the determinant.
    \(\left| {\begin{array}{*{20}{c}}
    {k{a_1}}&{k{b_1}}\\
    {{a_2}}&{{b_2}}
    \end{array}} \right| =\) \( k\left| {\begin{array}{*{20}{c}}
    {{a_1}}&{{b_1}}\\
    {{a_2}}&{{b_2}}
    \end{array}} \right|\)
  8. Linear combination of the elements of a determinant
    If the elements of any row (or column) multiplied by a constant factor are added to the corresponding elements of the other row (or column), then the value of the determinant does not change:
    \(\left| {\begin{array}{*{20}{c}}
    {{a_1} + k{b_1}}&{{b_1}}\\
    {{a_2} + k{b_2}}&{{b_2}}
    \end{array}} \right| =\) \( \left| {\begin{array}{*{20}{c}}
    {{a_1}}&{{b_1}}\\
    {{a_2}}&{{b_2}}
    \end{array}} \right|\)