# 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
$${\left( {AB} \right)^T} = {B^T}{A^T}$$.
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
$$\left| {A – \lambda I} \right| = 0$$.