Formulas and Tables

Matrices and Determinants

Properties of Matrices

Matrices: \(A\), \(B\), \(C\)
Elements of matrices: \({a_{ij}},\) \({b_{ij}},\) \({c_{ij}},\) \({b_i}\)
Identity matrix: \(I\)
Determinant of a matrix: \(\det A\)
Minor of an element \({a_{ij}}\): \({M_{ij}}\)
Cofactor of an element \({a_{ij}}\): \({A_{ij}}\)
Transpose of a matrix: \({A_T}\)

Adjoint of a matrix: \({C^*}\)
Inverse matrix: \({A^{-1}}\)
Trace of a matrix: \(\text{tr }A\)
Eigenvectors: \(X\)
Eigenvalues: \(\lambda\)
Real number: \(k\)
Natural numbers: \(m\), \(n\), \(i\), \(j\)

  1. Definition of a matrix
    An \(m \times n\) matrix \(A\) is a rectangular array of elements \({a_{ij}}\) (as a rule, these are numbers or functions), consisting of \(m\) rows and \(n\) columns.
    \(A = \left[ {{a_{ij}}} \right] =\) \( \left[ {\begin{array}{*{20}{c}}
    {{a_{11}}} & {{a_{12}}} & \ldots & {{a_{1n}}}\\
    {{a_{21}}} & {{a_{22}}} & \ldots & {{a_{2n}}}\\
    \vdots & \vdots & {} & \vdots \\
    {{a_{m1}}} & {{a_{m2}}} & \ldots & {{a_{mn}}}
    \end{array}} \right]\)
  2. A square matrix of order \(n\) has \(n\) rows and \(n\) columns.
  3. A square matrix \(\left[ {{a_{ij}}} \right]\) is called a symmetric matrix if \({{a_{ij}}} = {{a_{ji}}},\) i.e. the elements of the matrix are symmetric with respect to the main diagonal.
  4. A square matrix \(\left[ {{a_{ij}}} \right]\) is called skew-symmetric if \({{a_{ij}}} = -{{a_{ji}}}\).
  5. A square matrix is called diagonal if all its elements outside the main diagonal are equal to zero.
  6. A diagonal matrix is called the identity matrix if the elements on its main diagonal are all equal to \(1.\) (All other elements are zero).
  7. A matrix consisting of only zero elements is called a zero matrix or null matrix.
  8. Equality of matrices
    Two matrices \(A\) and \(B\) are equal if and only if they have the same size \(m \times n\) and their corresponding elements are equal.
  9. Addition and subtraction of matrices
    Two matrices \(A\) and \(B\) can be added (or subtracted) if and only if they have the same size \(m \times n\). If
    \(A = \left[ {{a_{ij}}} \right] =\) \( \left[ {\begin{array}{*{20}{c}}
    {{a_{11}}} & \ldots & {{a_{1n}}}\\
    {{a_{21}}} & \ldots & {{a_{2n}}}\\
    \vdots & {} & \vdots \\
    {{a_{m1}}} & \ldots & {{a_{mn}}}
    \end{array}} \right],\;\) \(B = \left[ {{b_{ij}}} \right] =\) \( \left[ {\begin{array}{*{20}{c}}
    {{b_{11}}} & \ldots & {{b_{1n}}}\\
    {{b_{21}}} & \ldots & {{b_{2n}}}\\
    \vdots & {} & \vdots \\
    {{b_{m1}}} & \ldots & {{b_{mn}}}
    \end{array}} \right],\)
    then the sum of these matrices is defined as
    \(A + B =\) \(\left[ {\begin{array}{*{20}{c}}
    {{a_{11}} + {b_{11}}}& \ldots &{{a_{1n}} + {b_{1n}}}\\
    {{a_{21}} + {b_{21}}}& \ldots &{{a_{2n}} + {b_{2n}}}\\
    \vdots & {}& \vdots \\
    {{a_{m1}} + {b_{m1}}}& \ldots &{{a_{mn}} + {b_{mn}}}
    \end{array}} \right]\)
  10. Multiplication of a matrix by a number
    Given a constant number \(k\) and a matrix \(A = \left[ {{a_{ij}}} \right]\). Then
    \(kA = \left[ {{ka_{ij}}} \right] =\) \( \left[ {\begin{array}{*{20}{c}}
    {{ka_{11}}} & {{ka_{12}}} & \ldots & {{ka_{1n}}}\\
    {{ka_{21}}} & {{ka_{22}}} & \ldots & {{ka_{2n}}}\\
    \vdots & \vdots & {} & \vdots \\
    {{ka_{m1}}} & {{ka_{m2}}} & \ldots & {{ka_{mn}}}
    \end{array}} \right].\)
  11. Matrix multiplication
    Let \(A\) and \(B\) be two matrices. The product of the matrices \(AB\) exists if and only if the number of columns in the first matrix is equal to the number of rows in the second matrix. If
    \(A = \left[ {{a_{ij}}} \right] =\) \( \left[ {\begin{array}{*{20}{c}}
    {{a_{11}}} & {{a_{12}}} & \ldots & {{a_{1n}}}\\
    {{a_{21}}} & {{a_{22}}} & \ldots & {{a_{2n}}}\\
    \vdots & \vdots & {} & \vdots \\
    {{a_{m1}}} & {{a_{m2}}} & \ldots & {{a_{mn}}}
    \end{array}} \right],\;\) \(B = \left[ {{b_{ij}}} \right] =\) \( \left[ {\begin{array}{*{20}{c}}
    {{b_{11}}} & {{b_{12}}} & \ldots & {{b_{1k}}}\\
    {{b_{21}}} & {{b_{22}}} & \ldots & {{b_{2k}}}\\
    \vdots & \vdots & {} & \vdots \\
    {{b_{n1}}} & {{b_{n2}}} & \ldots & {{b_{nk}}}
    \end{array}} \right],\)
    then the product \(AB\) is represented as a matrix
    \(AB = C =\) \( \left[ {\begin{array}{*{20}{c}}
    {{c_{11}}} & {{c_{12}}} & \ldots & {{c_{1k}}}\\
    {{c_{21}}} & {{c_{22}}} & \ldots & {{c_{2k}}}\\
    \vdots & \vdots & {} & \vdots \\
    {{c_{m1}}} & {{c_{m2}}} & \ldots & {{c_{mk}}}
    \end{array}} \right],\)
    where the elements of the matrix \(C\) are defined as
    \({c_{ij}} = {a_{i1}}{b_{1j}} + {a_{i2}}{b_{2j}} + \ldots \) \(+\; {a_{in}}{b_{nj}} = \sum\limits_{\lambda = 1}^n {{a_{i\lambda }}{b_{\lambda j}}},\;\) \(\big( {i = 1,2, \ldots ,m,}\;\) \({j = 1,2, \ldots ,k} \big)\)
    For example, if
    \(A = \left[ {{a_{ij}}} \right] =\) \( \left[ {\begin{array}{*{20}{c}}
    {{a_{11}}}&{{a_{12}}}&{{a_{13}}}\\
    {{a_{21}}}&{{a_{22}}}&{{a_{31}}}
    \end{array}} \right],\;\) \(B = \left[ {{b_i}} \right] = \left[ {\begin{array}{*{20}{c}}
    {{b_1}}\\
    {{b_2}}\\
    {{b_3}}
    \end{array}} \right],\)
    then the product \(AB\) is given by
    \(AB =\) \(\left[ {\begin{array}{*{20}{c}}
    {{a_{11}}}&{{a_{12}}}&{{a_{13}}}\\
    {{a_{21}}}&{{a_{22}}}&{{a_{31}}}
    \end{array}} \right] \cdot \left[ {\begin{array}{*{20}{c}}
    {{b_1}}\\
    {{b_2}}\\
    {{b_3}}
    \end{array}} \right] =\) \( \left[ {\begin{array}{*{20}{c}}
    {{c_{11}}}\\
    {{c_{21}}}
    \end{array}} \right] =\) \( \left[ {\begin{array}{*{20}{c}}
    {{a_{11}}{b_1} + {a_{12}}{b_2} + {a_{13}}{b_3}}\\
    {{a_{21}}{b_1} + {a_{22}}{b_2} + {a_{23}}{b_3}}
    \end{array}} \right].\)
  12. Transpose of a matrix
    If the rows and columns in a matrix \(A\) are interchanged, the new matrix is called the transpose of the original matrix \(A.\) The transposed matrix is denoted by \(A^T\).
  13. A square matrix \(A\) is called orthogonal if \(A{A^T} = I,\) where \(I\) is the identity matrix.
  14. If the matrix product \(AB\) is defined, then
    \({\left( {AB} \right)^T} = {B^T}{A^T}\).
  15. Adjoint of a matrix
    If \(A\) is a square matrix of order \(n\), then the corresponding adjoint matrix, denoted as \(C^*\), is a matrix formed by the cofactors \({A_{ij}}\) of the elements of the transposed matrix \(A^T\).
  16. Trace of a matrix
    If \(A\) is a square matrix of order \(n\), then its trace, denoted as \(\text{tr }A,\) is the sum of the elements on the main diagonal:
    \(\text{tr }A =\) \({a_{11}} + {a_{22}} + {a_{33}} + \ldots \) \(+\; {a_{nn}}.\)
  17. Inverse of a matrix
    The inverse of a matrix \(A\) is defined as a matrix \(A^{-1}\) such that the result of multiplication of the original matrix \(A\) by \(A^{-1}\) is the identity matrix \(I:\)
    \(A{A^{ – 1}} = I\).
    An inverse matrix exists only for square nonsingular matrices (whose determinant is not zero). If \(A\) is a square nonsingular matrix of order \(n,\) the inverse matrix \(A^{-1}\) is given by
    \({A^{ – 1}} = {\large\frac{{{C^*}}}{{\det A}}\normalsize},\)
    where \(C^*\) is the adjoint of the matrix and \(\det A\) is the determinant of the matrix \(A.\)
  18. If the matrix product \(AB\) is defined, then
    \({\left( {AB} \right)^{ – 1}} = {B^{ – 1}}{A^{ – 1}}\).
  19. Eigenvectors and eigenvalues of a matrix
    If \(A\) is a square matrix, its eigenvectors \(X\) satisfy the matrix equation
    \(AX = \lambda X\),
    and the eigenvalues \(\lambda\) are determined by the characteristic equation
    \(\left| {A – \lambda I} \right| = 0\).