SHORT NOTES: Matrices and Determinants

A rectangular array of 'mn' numbers in the form of m rows and n columns is called matrix of order
m by n written as m x n matrix

$\begin{bmatrix} a_{11} & a_{12} & a_{13} & ... &a_{1n} \\ a_{21}& a_{22} & a_{23} & ... &a_{2n} \\ ...&... &... &... &... \\ a_{m1}&a_{m2} &a_{m3} &... &a_{}mn \end{bmatrix}$

Equality of matrices:

Two matrices A and B are equal if they are of same order and all corresponding elements are equal.

Addition of Matrices:

$A \pm B = \left[ {{a_{ij}}} \right]\; \pm \;\left[ {{b_{ij}}} \right]$

For example

$\left[ {\;\begin{array}{*{20}{c}} 1&{ - 2}\\ 3&{\;\;4}\\ 5&{ - 3} \end{array}\;} \right]\;\; + \;\;\left[ {\begin{array}{*{20}{c}} {\;\;2}&{\;1}\\ { - 1}&{\;3}\\ {\;\;4}&{\;2} \end{array}\;} \right]\;\; = \;\left[ {\;\begin{array}{*{20}{c}} 3&{\; - 1}\\ 2&{\;\;\;7}\\ 9&{\; - 1} \end{array}\;} \right]\;$

Multiplication by a scalar:

If we multiply a matrix $A = [{a_{ij}}]$ by $\lambda ,$ we get the matrix $\lambda A = [\lambda {a_{ij}}].\;$

Example

$4\left[ {\;\begin{array}{*{20}{c}} 1&3&{\;\;2}\\ 2&1&{ - 1} \end{array}\;} \right]\;\; = \;\;\left[ {\;\begin{array}{*{20}{c}} 4&{12}&{\;\;8}\\ 8&4&{ - 4} \end{array}\;} \right]$

Multiplication Of Matrices:

$A = \left[ {\begin{array}{*{20}{c}} {{a_1}}&{{a_2}}&{{a_3}}\\ {{b_1}}&{{b_2}}&{{b_3}}\\ {{c_1}}&{{c_2}}&{{c_3}} \end{array}} \right] \,\,\,\,\,\, B = \left[ {\begin{array}{*{20}{c}} {{\alpha _1}}&{{\beta _1}}\\ {{\alpha _1}}&{{\beta _2}}\\ {{\alpha _3}}&{{\beta _3}} \end{array}} \right]$

To obtain the element ${a_{11}}$ of $AB$ , we multiply ${R_1}$ of $A$ with ${C_1}$ of $B$ :

To obtain the element ${a_{12}}$ of $AB,$ we multiply ${R_1}$ of $A$ with ${C_2}$ of $B$ :

To obtain the element ${a_{21}}$ of $AB,$ we multiply ${R_2}$ of $A$ with ${C_1}$ of $B$ :

Proceeding this way, we obtain all the elements of $AB$ .

If $A$ is or order $m \times n$ , and $B$ of order $n \times p$ , then to obtain the element ${a_{ij}}$ in $AB$ , we multiply ${R_i}$ in $A$ with ${C_j}$ in $B$ :

Transpose:

$A = \left[ {\;\begin{array}{*{20}{c}} 2&4\\ 1&1\\ 3&7 \end{array}\;} \right] \,\,\,\,\,\,\, \Rightarrow \,\,\,\, {A^T} = \left[ {\;\begin{array}{*{20}{c}} 2&1&3\\ 4&1&7 \end{array}\;} \right]$

Note:- $\left ( AB \right )^T=B^TA^T$ ,

$\left ( A+B \right )^T=A^T+B^T$

$\left (A^T \right )^T=A$

Determinant of Matrices:

$\Delta = \left| {\;\begin{array}{*{20}{c}} a\,\,\,\,b\\ c\,\,\,\,d \end{array}\;} \right|$

$\Delta = ad - bc.$

$\Delta = \left| {\;\begin{array}{*{20}{c}} {{a_1}}\,\,\,\,{{a_2}}\,\,\,\,{{a_3}}\\ {{b_1}}\,\,\,\,{{b_2}}\,\,\,\,{{b_3}}\\ {{c_1}}\,\,\,\,{{c_2}}\,\,\,\,{{c_3}} \end{array}\;} \right|$

$= {a_1}\left| {\begin{array}{*{20}{c}} {{b_2}}&{{b_3}}\\ {{c_2}}&{{c_3}} \end{array}} \right| - {a_2}\left| {\begin{array}{*{20}{c}} {{b_1}}&{{b_3}}\\ {{c_1}}&{{c_3}} \end{array}} \right| + {a_3}\left| {\begin{array}{*{20}{c}} {{b_1}}&{{b_2}}\\ {{c_1}}&{{c_2}} \end{array}} \right|$

$= {a_1}\left( {{b_2}{c_3} - {b_3}{c_2}} \right) - {a_2}\left( {{b_1}{c_3} - {b_3}{c_1}} \right) + {a_3}\left( {{b_1}{c_2} - {b_2}{c_1}} \right)$

$\Delta =a_1C_{11}+a_2C_{12}+a_3C_{13}$

${a_{ij}}$ that element which is in the ${i^{{\rm{th}}}}$ row and ${j^{{\rm{th}}}}$ column

Each ${a_{ij}}$ will have its corresponding co-factor ${C_{ij}}$

${C_{ij}} = {\left( { - 1} \right)^{i + j}}{M_{ij}}$

$M_{ij}$ is called minor of ${a_{ij}}$ which is calculated by taking the determinant of matrix formed by

eleminating the 'i' row and 'j' column.

For example minor of $a_{1}$ is $M_{11}=\begin{vmatrix} b_2 & b_3\\ c_2& c_3 \end{vmatrix}=b_2c_3-b_3c_2$

Properties of determinant:

The value of determinant is not changed when rows are changed into columns and columns into rows.
If any two rows or columns of a determinant are interchanged, the sign of the determinant changes but its magnitude remains the same
A determinant having two rows or two columns identical has the value zero
Multiplying all the elements of a row (or column) by a scalar (a real number) is equivalent to multiplying the determinant by that scalar
A determinant can be split into a sum of two determinants along any row or column

Inverse of a square matrix :

Order two :

$AA^{-1}=I\\A=\begin{bmatrix} a &b \\ c& d \end{bmatrix}\\ A^{-1}=\frac{1}{ad-bc}\begin{bmatrix} d &-b \\ -c&a \end{bmatrix}$

Order three :

$A = \left[ {\;\begin{array}{*{20}{c}} {{a_1}}&{{a_2}}&{{a_3}}\\ {{a_4}}&{{a_5}}&{{a_6}}\\ {{a_7}}&{{a_8}}&{{a_9}} \end{array}\;} \right]$ with corresponding co-factors ${C_1},{C_2}\ldots {C_9}$

$\widetilde A = \left[ {\;\begin{array}{*{20}{c}} {{C_1}}&{{C_4}}&{{C_7}}\\ {{C_2}}&{{C_5}}&{{C_8}}\\ {{C_3}}&{{C_6}}&{{C_9}} \end{array}\;} \right]$ Adjoint of A=Transpose of cofactor matrix

${\color{Magenta} A^{-1}=\frac{1}{\left | A \right |}\left ( {\color{Blue} adj} A \right )}$

Note:

${\color{Blue} \left ( A^T \right )^{-1}=\left ( A^{-1} \right )^T}\\ {\color{Cyan} \left ( AB \right )^{-1}=B^{-1}A^{-1}}$

Diagonal matrix All non-diagonal elements are zero.

$\left[ {\;\begin{array}{*{20}{c}} 1&0&0\\ 0&{ - 1}&0\\ 0&0&2 \end{array}\;} \right]$

Symmetric matrix:

$A^T=A$ i.e $a_{ij}=a_{ji}$

$A=\begin{bmatrix} 2 &1 &5 \\ 1& 0 &-3 \\ 5& -3& 6 \end{bmatrix}$

Skew Symmetric Matrix:

$A^T=-A$ i.e $a_{ii}=0,a_{ij}=-a_{ji}$

Main diagonal elements are zero

$A=\begin{bmatrix} 0 &1 &5 \\ -1& 0 &-3 \\ -5& 3& 0 \end{bmatrix}$

Solutions of simultaneous linear equations:

1.Cramer's rule:

Consider a system

$a_1x+b_1y+c_1z=d_1\\ a_2x+b_2y+c_2z=d_2\\ a_3x+b_3y+c_3z=d_3$

$D=\begin{vmatrix} a_1 & b_1& c_1\\ a_2& b_2 & c_2\\ a_3& b_3 & c_3 \end{vmatrix}$

$D_1=\begin{vmatrix} d_1 & b_1& c_1\\ d_2& b_2 & c_2\\ d_3& b_3 & c_3 \end{vmatrix}$ $D_2=\begin{vmatrix} a_1 & d_1& c_1\\ a_2& d_2 & c_2\\ a_3& d_3 & c_3 \end{vmatrix}$ $D_3=\begin{vmatrix} a_1 & b_1& d_1\\ a_2& b_2 & d_2\\ a_3& b_3 & d_3 \end{vmatrix}$

${\color{Blue} x=\frac{D_1}{D},y=\frac{D_2}{D},z=\frac{D_3}{D}}$

$D\neq0$ consistent and unique solution
$D=0$ and at least one of $D_1,D_2,D_3$ is non zero-Inconsistent and no solution
$D=D_1=D_2=D_3=0$ infinitely many solution

2.Matrix method:

$a_1x+b_1y+c_1z=d_1\\ a_2x+b_2y+c_2z=d_2\\ a_3x+b_3y+c_3z=d_3$

${\color{Magenta} \Rightarrow AX=B\Rightarrow X=A^{-1}B}$

ELECTRICAL ENGINEERING
ENGINEERING MATHEMATICS	Electric Circuits and Fields
Signals and Systems	Electrical Machines
Power Systems	Control Systems
Electrical and Electronic Measurements	Power Electronics and Drives
Analog Electronics	Digital Electronics

Matrices and Determinants

No comments:

Post a Comment