Matrics and Determinants

I. A set of minimum numbers (real or complex), arrange in a rectangular array of 'm' rows and colums is called a matrix of order m x n

ie. A=${[a_{\mathrm{ij}}]}_{mxn}$ where i=1,2,3.....m and j= 1,2,3..... n.

Types of Matrices

1. The matrix A = ${[a_{\mathrm{ij}}]}_{mxn}$ is said to be a

• Rectangular Matrix of order m x n if m$\ne n$
• Square matrix of order n, if m=n
• Row matrix (row vector), if m=1
• Column Matrix (Colum vector) if n=1
• Zero Matrix (null matrix), if $a_{\mathrm{ij}}$ =0 for all ij

In a square matrix A= [$a_{\mathrm{ij}}$] of order 'n' for i-j are diagonal elements of A. The sum of the diagonal elemements is called Trace of A.

Thus Trace of A= Tr(A)= $a_{11}$+$a_{22}$+$a_{33}$+---------+$a_{\mathrm{nn}}$

II. The square matrix A= [$a_{\mathrm{ij}}$] of order 'n' is said to be

1. Upper Triangular if $a_{\mathrm{ij}}$= 0 for all i >j.
2. Lower triangular if $a_{\mathrm{ij}}$=0 for i<j.
3. The triangular matrix, if it is either upper triangular or lower triangular or both.

III. The square matrix A=${[a}_{\mathrm{ij}}$] of order'n' is said to be

1. diagonal matrix if $a_{\mathrm{ij}}$=0 for all i$\ne j$
2. Scalar matrix if $a_{\mathrm{ij}}$=0 for all i$\ne$ j and $a_{\mathrm{ij}}$=k for all i=j
3. Unit matrix (identity matrix) if $a_{\mathrm{ij}}$= 0 for all i$\ne$j and $a_{\mathrm{ij}}$=1 for all i=j

A unit matrix of order 'n' is denoted by $I_n$or I

IV. Algebra of Matrix

(1)Equality of matrices:

Two matrices A and B are said to be equal if they are of the same order and their corresponding elements are also equal.

(2) Addition of Matrics

If A= ${[a_{\mathrm{ij}}]}_{\mathrm{mxn}}$ and B= ${[b_{\mathrm{ij}}]}_{mxn}$then

(a) sum of A+B= ${[a_{\mathrm{ij}}+b_{\mathrm{ij}}]}_{m\mathrm{xn}}$

(b) Difference A-B= ${[a_{\mathrm{ij}}-b_{\mathrm{ij}}]}_{mxn}$

Properties of Matrix addtion;

If A,B,C are three matrices of the same order, then

(a) A+B=B+A(Commutative)

(b)(A+B)+C=A+(B+C)(Associative)

(c)Corresponding to every matrixA, there exists a zero matrix 'O' fo the same order such cthat

A+O=A=O+A then O is the additive identity of A

(d)A+(-A)=O=(-A)+A where -A is the additive inverse of A.

(e)A+B=A+C$\Rightarrow$B=C.

(3) Multiplication of a matrix by a scalar

If A is any given matrixand k be a scalar, then kA is the matrix obtained by multiplying every elements of A by k.

Properties of Scalar Multiplication

If A and B are two matrices of the same order, then

a) ($k_1$+$k_2$)A=$k_1$A+$k_2$A

b)$k_1$($k_2$A)=($k_1$ $k_2$)A

c)k(A+B)=kA+kB

d)(-kA)=-(kA)=k(-A)

4. Multiplication of Matrices

If A ${[a_{\mathrm{ij}}]}_{mxn}$ and B=${[b_{\mathrm{ij}}]}_{nxp}$ be two matrices such that the number of colums of A is equal to the number of rows of B, then the product AB is defined as the matix C=${[c_{\mathrm{ij}}]}_{mxp}$ of order m x p

where $C_{\mathrm{ij}}$= $a_{i1}{b}_{1j}$+ $a_{i2}{b}_{2j}$+$a_{i3}$$b_{3j}$+............+ $a_{\mathrm{ip}}$$b_{\mathrm{pj}}$

=$\underset{k=1}{\overset{p}{\mathrm{\Sigma }}}$ $a_{\mathrm{ik}}$ $b_{\mathrm{kj}}$ for i=1, 2, 3.. m and j= 1,2, 3...m

The product AB is defined iff the number of columns of A is equal to the number of rows of B. Two such matrices are said to be comfortable for multiplication.

Properties of matrix multiplication

A, B, C are matrices comfortable for the indicated signs and products, then

a) AB$\ne \mathrm{BA}$(multiplication is not commutative)

b)A(BC)=(AB)C (Associative)

c) A (B+C)= AB+ BC(right distributive)

e) AO=0 and OA=o.

f)AB=0 does not necessarilyimply that A=0 or B=0 or both are zeros.

g)AB=AC need not imply B=c

h) If A is a square matrix of order n,

Then $I_n$.A=A.$I_{n\mathrm{.}}$

If A and B are square matrices of the same order, then

a) ${(A+B)}^2$=$A^2$+ AB+ BA+ $B^2$

=$A^2$+2Ab+$b^2$, Iff AB= BA

b)${(A-B)}^2$=$A^2$-AB-BA+$B^2$, iff AB=BA

c)(A+B)(A-B)=$A^2$-AB-BA-$B^2$

=$A^2$-$B^2$, iff AB-BA

A square matrix A is called

(1) an idempotent Matrix if $A^2$=A

(2) a nilpotent matrix if $A^k$=0 where k is a positive integer. If k is the least positive integer such that $A^k$=0 where K is a positive integer. If k is the least positive integer such that $A^k$=0, then k is called the index of the nilpotent matrix A.

(3) An involutory matrix if $A^2$=I

note that unit matrix is involutory.

V. Related Matrices

1. The transpose of a matrix A=${[C_{\mathrm{ij}}]}_{mxn}$ of order m x n is an n x m matrix A' r $A^T$= ${{[a}_{\mathrm{ij}}]}_{mxn}$ the matrix obtained by interchanging the rows and columns of A.

Properties of the transpose

a) ${{(A}^T)}^T$=A

b)${(\mathrm{kA})}^T$=K.$A^T$ , k be any scalar

c)${(A+B)}^T$ = $A^T$-$B^T$

e)${(\mathrm{AB})}^T$=$B^T$ $A^T$ , if A and B are comfortable for multiplication.

f) $\left|A^T\right|$ = $\left|A\right|$

2. Conjugate of a matrix A = ${{[a}_{\mathrm{ij}}]}_{mxn}$ where $\overline{A}$=${{[a}_{\mathrm{ij}}]}_{mxn}$. where $\overline{a_{\mathrm{ij}}}$ is the complex conjugate of a $a_{\mathrm{ij}}$. The matrix obtained by replacing all the elements by their corresponding compelex conjugate is called the conjugate of A, denoted by $\overline{A}$.

Properties of the conjugate

(a) $\overline{(\overline{A)}}$= A

(b) $\overline{\mathrm{kA}}$= K.$\overline{A}$

(c) $\overline{(\overline{A\pm B}}$)= A$\pm B$

(d)$\overline{\mathrm{AB}}$= $\overline{B}$.$\overline{A}$

(e)${(\overline{A)}}^T$=$\overline{(A^T)}$= $A^{\ast }$

Transpose of the magnitude of A is the conjugate of the transpose of A and is denoted by '$A^{\ast }$' or '$A^{\theta }$'

Properties of Conjugate transpose

(a) ${(A^{\ast })}^{\ast }$=A

(b)${(\mathrm{kA})}^{\ast }$= ${\mathrm{kA}}^{\ast }$

(c) ${(A\pm B)}^{\ast }$=$A^{\ast }\pm B^{\ast }$

(d) ${(\mathrm{AB})}^{\ast }$=$B^{\ast }$.$A^{\ast }$

(e) |$A^{\ast }$| = |$\overline{A}$|

3. A square matrix A= [$a_{\mathrm{ij}}$] of order 'n' is said to be,
(a)Symmetric, If $A^T$ = A, ie $a_{\mathrm{ij}}$= $a_{\mathrm{ji}}$ for all i and J

(b) Skew -Symmetric, If $A^T$= -A, ie, $a_{\mathrm{ji}}$= -$a_{\mathrm{ij}}$ for all i & J

Note 1

1) All the diagonal elements of a skew symmetric matrix are zeros.

2) Determinant of a skew -symmetric matrix of odd order is zero and that of even order is a perfect square.

3) Hermitian, if $A^{\ast }$ = A, ie $\overline{a_{\mathrm{ij}}}$- $\overline{a_{\mathrm{ji}}}$ for all i and j

4) Skew- hermitian if $A^{\ast }$= -A ie, $\overline{a_{\mathrm{ij}}}$= -$\overline{a_{\mathrm{ij}}}$ for all i and j

Note 2.

1) All the diagonal elements of a hermitian matrix are real numbers

2)All the diagonal elements of a skew hermitian matrix are purely imaginary or zeros.

3)Determinant of a hermitian matrix is always a real number.

4)Every square matrix A can be expressed uniquely as the sum of a symmetric matrix and a skew -symmetric matrix ie, A = P+ Q where P is symmetric and Q is skew symmetric such that P= $\frac{1}{2}$(A+$A^T$) and Q= $\frac{1}{2}$(A-$A^T$)

5. Every square matrix A can be uniquely expressed as the sum f a hermitian matrix and a skwe -hermiatian matrix. ie, A= L+ M, where L is hermitian and M is skew - hermitian such that L = $\frac{1}{2}$(A+$A^{\ast }$) adn M= $\frac{1}{2}$(A-$A^{\ast }$)

(a) Every real Symmetric matrix is hermtian.

(b)Every real skew Symmetric matrix is skew -hermitian.

6. If A and B are symmetric matrices fo the same order then AB+ BA is symmetric and AB- BA is skew symmetric.

7 A square matrix A is said to be

(a) singular if |A|= 0

(b) non-singular, if |A|$\ne 0.$

Minor Co-Factors

The minor of an element of a square matrix is denoted by $M_{\mathrm{ij}}$ by deleating the $i^{\mathrm{th}}$ row and $j^{\mathrm{th}}$ column.

The co-factor elemetn $a_{\mathrm{ij}}$ is denoted by $c_{\mathrm{ij}}$= ${(-1)}^{i+j}$ $M_{\mathrm{ij}}$ matrix of A and is denoted by 'adjA' . if A is any square matrix of order n, then

a)A(adj A)= |A|. $I_n$= (adj A). A

b)|A(AdjA)|= ${|A|}^n$

c)If A is a non -singular, then

1) $\frac{A(\mathrm{adj}A)}{|A|}$= $I_n$= $\frac{(\mathrm{adj}A)A}{|A|}$

2) |adj A|= ${|A|}^{n-1}$

3)adj (adj A)= ${|A|}^{n-2}$.A is a square matrix of order 2.

d) If A and B are the square matrices of the same order then adj(AB) =(adj B). (adj A)

9. Inverse of a Matrix

Let A be a square matrix of order N if there exists a matrix b such that AB= $I_n$= BA, where $I_n$ is the unit matrix of order n, then B is called the multiplicative inverse of A and is denoted by '$A^{-1}$

Properties of Inverse

1) A square matrix A is invertible iff A is non singular , (ie |A|$\ne$0)

2) An invertible matrix A has unique inverse.

$A^{-1}$= $\frac{\mathrm{adj}A}{|A|}$

3) If A and B are invertible, then AB is also invertible and ${(\mathrm{AB})}^{-1}$= $B^{-1}$. $A^{-1}$

4) If A is non singular |$A^{-1}$|= $\frac{1}{|A|}$

5) If A =$\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ , then $A^{-1}$=$\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]$, if ad-bc$\ne 0$

10. A square matrix is said to be

1) orthogonal if $A^T$A= I = A$A^T$

Note 3

a) For an orthogonal matrix |A|=$\pm$1

b)Inverse of an orthoganl matrix is also orthogonal

c)the product of two orthogonal matrix is orthogonal

2. A is a unitary matrix if $A^{\ast }$A= I =A$A^{\ast }$

a)For unitary matrix A; |A|= I

b)The inverse of a unitary matrix A is $A^{-1}$= A

Determinants

1. Determinants of a square matrix A = [a] of order 1 is |A|= |a|= a

2. Determinant of a square matrix A = $\left[\begin{array}{ccc}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right]$ of order 3 is |A| = $\left[\begin{array}{ccc}{a}_1& b_1& c_1\\ a_2& b_2& b_2\\ a_3& b_3& c_3\end{array}\right]$=$\mathrm{\Sigma }{a}_1$($b_2{c}_3$-$b_3{c}_2$)

Properties of Determinants

1. The Value of a determinant is unaltered if its rows and columns are interchanged id $|A^T$|= |A|
2. If any two rows or columns of a determinant are interchanged, then the sign of the determinant changes.
3. If ay two rows or columns or a determinant are identical (or propotional) then the determinant vanishes.
4. If evey element of a row or column of a determinant is multiplied by k, then its value gets multiplied by k
   If A is square matrix of order 'n' then |kA| = $K^{n\mathrm{.}}$|A|.
5. If each element of row or column of a determinant is a sum of two or more terms then the determinant can be expressed as the sum of two or more determinants.
6. If each element of any row or column of a determinantis multiplied by any non-zero number and added to the corresponding elements of any other row or column, the value of the determinant is unaltered.
7. If A and B are two square matrices of the same order then |AB|= |A|. |B|.

Differentiation of a determinant:

1) If $\mathrm{\Delta }(x)$= |$c_1$ $c_2$ $c_3$| or $\left|\begin{array}{c}{R}_1\\ R_2\\ R_3\end{array}\right|$ , then

$\frac{d}{\mathrm{dx}}\mathrm{\Delta }(x)$= |$c_1$$\text{'}{c}_2$$c_3$| + |$c_1$$c_2\text{'}$$c_3$|+|$c_1$$c_2$$c_3\text{'}$|

or $\left|\begin{array}{c}{R}_1\text{'}\\ R_2\\ R_3\end{array}\right|+\left|\begin{array}{c}{R}_1\\ R_2\text{'}\\ R_3\end{array}\right|+\left|\begin{array}{c}{R}_1\\ R_2\\ R_3\text{'}\end{array}\right|$

2) if $\mathrm{\Delta }(x)$= $\left|\begin{array}{ccc}f(x)& g(x)& h(x)\\ b_1& b_2& b_3\\ c_1& c_2& c_3\end{array}\right|$ consists of a function of x in one row only and the other rows are constants then

${\mathrm{\Delta }}^n$ (x)=$\left|\begin{array}{ccc}{\mathrm{\Delta }}^{n}f(x)& {\mathrm{\Delta }}^{n}g(x)& {\mathrm{\Delta }}^{n}h(x)\\ b_1& b_2& b_3\\ c_1& c_2& c_3\end{array}\right|$

System of Linear Equations

In the system AX = B of equations, the matrix A is called the coefficient matrix, X is the coloumn matrix of the variable and B is the column matrix of the constant.

The system AX= B of eqautions is said to be

1. Non=homogeneous if B$\ne 0$
2. Homogenous if B= 0
3. Consistent if it has one or more solutions
4. Incosistent if it has no solution

The system of non-homogeneous linear equations AX= B (B$\ne$0) is said to be

1. Consistent with unique solution if |A|$\ne$0 and the solution is X=$A^{-1}$ B.
2. Inconsistent (id has no solution) if |A|= 0 and (adj A )B is a non null matrix.
3. Consistent with infinite solution if |A| = o and (adj A) is a null matrix

The system of homogeneous linear equations AX= 0 where A is a square matrix is always consistent and

1. If |A| = 0 the system has only a trival solution; $x_1$=0, $x_2$=0 $x_3$=0... $x_n$=0
2. If |A|$\ne$ 0 then the system has infinitely many solutions (non-trival)

Cramer's Rule

The system of linear equations

$\begin{array}{c}{a}_{1}X+b_{1}Y+c_{1}Z=d_1\\ a_{2}X+b_{2}Y+c_{2}Z=d_2\\ a_{3}X+b_{3}Y+c_{3}Z=d_3\end{array}\}\Rightarrow$ $\left[\begin{array}{ccc}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]$= $\left[\begin{array}{c}{d}_1\\ d_2\\ d_3\end{array}\right]$ $\Rightarrow$ AX= B

Where A= $\left[\begin{array}{ccc}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right]$; X= $\left[\begin{array}{c}x\\ y\\ z\end{array}\right]$ , B= $\left[\begin{array}{c}{d}_1\\ d_2\\ d_3\end{array}\right]$

If |A|$\ne$0 then x= $\frac{|A_1|}{|A|}$; Y= $\frac{|A_2|}{|A|}$ ; Z= $\frac{|A_3|}{|A|}$

Where $A_1$= $\left[\begin{array}{ccc}{d}_1& b_1& c_1\\ d_2& b_2& c_2\\ d_3& b_3& c_3\end{array}\right]$ , $A_2$= $\left[\begin{array}{ccc}{a}_1& d_1& c_1\\ a_2& d_2& c_2\\ a_3& d_3& c_3\end{array}\right]$

$A_3$ = $\left[\begin{array}{ccc}{a}_1& b_1& d_1\\ a_2& b_2& d_2\\ a_3& b_3& d_3\end{array}\right]$

The system of the three linear equations in two variables

$\begin{array}{c}{a}_{1}x+b_{1}y+c_1=0\\ a_{2}x+b_{2}y+c_2=0\\ a_{3}x+b_{3}y+c_3=0\end{array}$ has a unique solution if

$\left|\begin{array}{ccc}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right|=0$