MATRICES

 MATRICES  - 

A matrix is an ordered rectangular array of numbers or functions. The numbers or functions are called the elements or the entries of matrix. An m x n matrix is an arrangement of mn objects (nit necessarily distinct) in m rows and n columns in the form 

In this case we can say, a matrix of order m x n or we can say m by n.Order of matrix is nothing it is m x n matrix for example 2 x 4 matrix.

The objects a₁₁,a₁₂,a₁₃  and so on.......a_{mn are called the elements of the matrix.Each element of the matrix can be real or complex number or a function of one or more variables or any other object.The element} [a_ij] which is common its called its general element.

If all the elements of matrix are real then it is called real matrix.

If all the elements of matrix are complex then it is called complex matrix.

The matrix are usually denoted by bold face upper face letters A,B,C,.... so on we can siply write a matrix as A= [a_ij]_{m x n}

_{TYPES OF MATRICES}

(i) Column matrix  or Column Vector - A matrix is said to be a column matrix if it has only one column. In general, A = [aij] m × 1 is a column matrix of order m × 1. 

(ii) Row matrix or Row Vector - A matrix is said to be a row matrix if it has only one row. In general, B = [bij] 1 × n is a row matrix of order 1 × n. 

(iii) Square matrix - A matrix in which the number of rows are equal to the number of columns, is said to be a square matrix. Thus an m × n matrix is said to be a square matrix if m = n and is known as a square matrix of order ‘n’.  In general, A = [aij] m × m is a square matrix of order m.

(iv) Rectangular matrix  - A matrix of order m x n where m ≠ n is called the rectangular matrix.

(v) Diagonal matrix - A square matrix B = [bij] m × m is said to be a diagonal matrix if all its non diagonal elements are zero, that is a matrix B = [bij] m × m is said to be a diagonal matrix if bij = 0, when i ≠ j. 

(vi) Scalar matrix - A diagonal matrix is said to be a scalar matrix if its diagonal elements are equal, that is, a square matrix B = [bij] n × n is said to be a scalar matrix if bij = 0, when i ≠ j bij = k, when i = j, for some constant k. 

 (vii) Identity matrix -  A square matrix in which elements in the diagonal are all 1 and rest are all zero is called an identity matrix. We can say that a scalar matrix is an identity matrix when k = 1. But every identity matrix is clearly a scalar matrix.

(viii) Zero matrix or null matrix - A matrix is said to be zero matrix or null matrix if all its elements are zero. For example, [0],  [0, 0] are all zero matrices. We denote zero matrix by O. Its order will be clear from the context.

(ix) Equal matrix - Two matrices are equal if their order is equal and their corresponding elements are equal. If A = [aij] m × n and B = [bij] p x q  then we can say that A = B if  

(a) m =p and n = q

(b) [aij] = [bij] for all ij

(x) Sub-matrix - A matrix obtained by omitting by some rows or columns from a given matrix A is called a sub-matrix of A. As convention of the given matrix is also taken as a sub-matrix of A.

(xi) Nilpotent matrix - Define A^m =  A X A X A X …………X K times if A^k .A is nilpotent of index 'K'.

(xii) Idempotent matrix - A matrix which, when multiplied by itself, yields itself. That is, the matrix is idempotent if and only if . For this product to be defined, must necessarily be a square matrix.  A² = A X A ; A² =A.

(xiii) Symmetric and skew-symmetric matrix -  A symmetric matrix and skew-symmetric matrix both are square matrices. But the difference between them is, the symmetric matrix is equal to its transpose whereas skew-symmetric matrix is a matrix whose transpose is equal to its negative. 

If A is a symmetric matrix, then A = AT  and if A is a skew-symmetric matrix then AT = – A.

Sum of symmetric and skew-symmetric matrix is A = 1/2 (A + A′) + 1/2 (A − A′)

(xiv) Triangular matrix - In the mathematical discipline of linear algebra, a triangular matrix is a special kind of square matrix. A square matrix is called lower triangular if all the entries above the main diagonal are zero. Similarly, a square matrix is called upper triangular if all the entries below the main diagonal are zero.

(xv) Conjugate matrix - The complex conjugate transpose of a matrix interchanges the row and column index for each element, reflecting the elements across the main diagonal. The operation also negates the imaginary part of any complex numbers. For example, if B = A' and A(1,2) is 1+1i , then the element B(2,1) is 1-1i .

(xvi) Hermitian and skew hermition matrix - An Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the i-th row and j-th column is equal to the complex conjugate of the element in the j-th row and i-th column, for all indices i and j:

${\displaystyle A{\text{ Hermitian}}\quad \iff \quad a_{ij}={\overline {{a}_{ji}}}}$ or in matrix form:

Hermitian matrices can be understood as the complex extension of real symmetric matrices.

If the conjugate transpose of a matrix ${\displaystyle A}$ is denoted by ${\displaystyle A^{\mathsf {H}}}$, then the Hermitian property can be written concisely as

${\displaystyle A{\text{ Hermitian}}\quad \iff \quad A=A^{\mathsf {H}}}$

In linear algebra, a square matrix with complex entries is said to be skew-Hermitian or anti-Hermitian if its conjugate transpose is the negative of the original matrix..That is, the matrix ${\displaystyle A}$ is skew-Hermitian if it satisfies the relation

${\displaystyle A{\text{ skew-Hermitian}}\quad \iff \quad A^{\mathsf {H}}=-A}$ 

where  denotes the conjugate transpose of the matrix . In component form, this means that 

for all indices  and , where  is the element in the -th row and -th column of , and the overline denotes complex conjugation.

sum of hermitian and skew hermition matrix A = 1/2 (A + A^T) + 1/2 (A − A^T)

 watch explain defination on youtube 

https://www.youtube.com/watch?v=43vfdZ-0GZc

• The system of equation in which the determinant of the coefficient is zero is called non-trivial solution. And the system of equation in which the determinant of the coefficient matrix is not zero but the solution are x=y=z=0 is called trivial solution.

Operations on Matrices

Addition, subtraction and multiplication are the basic operations on the matrix. To add or subtract matrices, these must be of identical order and for multiplication, the number of columns in the first matrix equals the number of rows in the second matrix.

• Addition of Matrices
• Subtraction of Matrices
• Scalar Multiplication of Matrices
• Multiplication of Matrices

Addition of Matrices

If A[aij]mxn and B[bij]mxn are two matrices of the same order then their sum A + B is a matrix, and each element of that matrix is the sum of the corresponding elements. i.e. A + B = [aij + bij]mxn

Consider the two matrices A & B of order 2 x 2. Then the sum is given by:

$\begin{bmatrix} a1 &b1 \\ c1 &d1 \end{bmatrix} + \begin{bmatrix} a2&b2 \\ c2& d2 \end{bmatrix} = \begin{bmatrix} a1+a2 &\ b1+b2 \\ c1+c2 &\ d1+d2 \end{bmatrix}$

Properties of Matrix Addition: If a, B and C are matrices of same order, then

(a) Commutative Law: A + B = B + A

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

(c) Identity of the Matrix: A + O =  O + A = A, where O is zero matrix which is additive identity of the matrix,

(d) Additive Inverse: A + (-A) = 0 = (-A) + A, where (-A) is obtained by changing the sign of every element of A which is additive inverse of the matrix,

(e) $\left. \begin{matrix} A+B=A+C \\ B+A=C+A \\ \end{matrix} \right\}\Rightarrow B=C$

(f) $tr\left( A\pm B \right)=tr\left( A \right)\pm tr\left( B \right)$

(g) If A + B = 0 = B + A, then B is called additive inverse of A and also A is called the additive inverse of A.

Subtraction of Matrices

If A and B are two matrices of the same order, then we define $A-B=A+\left( -B \right).$

Consider the two matrices A & B of order 2 x 2. Then the difference is given by:

$\begin{bmatrix} a1 &b1 \\ c1 &d1 \end{bmatrix} – \begin{bmatrix} a2&b2 \\ c2& d2 \end{bmatrix} = \begin{bmatrix} a1-a2 &\ b1-b2 \\ c1-c2 &\ d1-d2 \end{bmatrix}$

We can subtract the matrices by subtracting each element of one matrix from the corresponding element of the second matrix. i.e. A – B = [aij – bij]mxn

Scalar Multiplication of Matrices

If $A={{\left[ {{a}_{ij}} \right]}_{m\times n}}$m×n​ is a matrix and k any number, then the matrix which is obtained by multiplying the elements of A by k is called the scalar multiplication of A by k and it is denoted by k A thus if $A={{\left[ {{a}_{ij}} \right]}_{m\times n}}$m×n​

Then $k{{A}_{m\,\times n}}={{A}_{m\,\times \,n}}k=\left[ k{{a}_{i\times j}} \right]$

Properties of Scalar Multiplication: If A, B are matrices of the same order and λ and μ are any two scalars then;

(a) $\lambda \left( A+B \right)=\lambda A+\lambda B$

(b) $\left( \lambda +\mu \right)A=\lambda A+\mu A$

(c) $\lambda \left( \mu A \right)=\left( \lambda \,\mu A \right)=\mu \left( \lambda A \right)$

(d) $\left( -\lambda A \right)=-\left( \lambda A \right)=\lambda \left( -A \right)$

(e) $tr\left( kA \right)=k\,\,tr\,\,\left( A \right)$

Multiplication of Matrices

If A and B be any two matrices, then their product AB will be defined only when the number of columns in A is equal to the number of rows in B.

If $A={{\left[ {{a}_{ij}} \right]}_{m\,\times n}}. and \,B={{\left[ {{b}_{ij}} \right]}_{n\,\times p}} then\; their\; product\ AB=C={{\left[ {{c}_{ij}} \right]}_{m\,\times p}}$m×n​.andB=[bij​]n×p​thentheirproduct AB=C=[cij​]m×p​ will be a matrix of order m×p where ${{\left( AB \right)}_{ij}}={{C}_{ij}}=\sum\limits_{r=1}^{n}{{{a}_{ir}}{{b}_{rj}}}$

Properties of matrix multiplication

(a) Matrix multiplication is not commutative in general, i.e. in general $AB\ne BA.$

(b) Matrix multiplication is associative, i.e. (AB)C = A(BC).

(c) Matrix multiplication is distributive over matrix addition, i.e. A.(B + C) = A.B + A.C and (A + B)C = AC + BC.

(d) If A is an m × n matrix, then ${{I}_{m}}A=A=A{{I}_{n}}.$

(e) The product of two matrices can be a null matrix while neither of them is null, i.e. if AB = 0, it is not necessary that either A = 0 or B = 0.

(f) If A is an m × n matrix and O is a null matrix then ${{A}_{m\,\times n}}.{{O}_{n\,\times p}}={{O}_{m\,\times p}}.$ i.e. the product of the matrix with a null matrix is always a null matrix.

(g) If AB = 0 (It does not mean that A = 0 or B = 0, again the product of two non-zero matrices may be a zero matrix).

(h) If AB = AC , B ≠ C (Cancellation Law is not applicable).

(i) $tr\left( AB \right)=tr\left( BA \right).$

j) There exist a multiplicative identity for every square matrix such AI = IA = A

Matrix Operations Examples

Illustration 1: If $A=\left[ \begin{matrix} 2 & 1 & 3 \\ 3 & -2 & 1 \\ -1 & 0 & 1 \\ \end{matrix} \right] and B=\left[ \begin{matrix} 1 \\ 2 \\ 4 \\ \end{matrix}\,\,\,\,\begin{matrix} -2 \\ 1 \\ -2 \\ \end{matrix} \right]$

find AB and BA if possible.

Solution:

Using matrix multiplication. Here, A is a 3 × 3 matrix and B is a 3 × 2 matrix, therefore, A and B are conformable for the product AB and it is of the order 3 × 2 such that

(First row of A) (First column of B) $=\left[ 2\,1\,3 \right]\left[ \begin{matrix} 1 \\ 2 \\ 4 \\ \end{matrix} \right]=2\times 1+1\times 2+3\times 4=16$.

(First row of A) (Second column of B) $=\left[ 2\,\,1\,\,3 \right]\left[ \begin{matrix} -2 \\ 1 \\ -3 \\ \end{matrix} \right]=2\times \left( -2 \right)+1\times 1+3\times \left( -3 \right)=-12$

(Second row of A) (First column of B) $=\left[ 3\,\,-2\,\,1 \right]\left[ \begin{matrix} 1 \\ 2 \\ 4 \\ \end{matrix} \right]=3\times 1+\left( -2 \right)\times 2+1\times 4=3$

Similarly ${{\left( AB \right)}_{22}}=-10,{{\left( AB \right)}_{31}}=3 \;and \;{{\left( AB \right)}_{32}}=0$

∴ AB = $\left[ \begin{matrix} 16 \\ 3 \\ 3 \\ \end{matrix}\,\,\,\begin{matrix} -12 \\ -10 \\ 0 \\ \end{matrix} \right]$

BA is not possible since number of columns of $B\ne$ number of rows of A.

Illustration 2: Find the value of x and y if $2\left[ \begin{matrix} 1 & 3 \\ 0 & x \\ \end{matrix} \right]+\left[ \begin{matrix} y & 0 \\ 1 & 2 \\ \end{matrix} \right]=\left[ \begin{matrix} 5 & 6 \\ 1 & 8 \\ \end{matrix} \right]$

Solution:

Using the method of multiplication and addition of matrices, then equating the corresponding elements of L.H.S. and R.H.S., we can easily get the required values of x and y.

We have, $2\left[ \begin{matrix} 1 & 3 \\ 0 & x \\ \end{matrix} \right]+\left[ \begin{matrix} y & 0 \\ 1 & 2 \\ \end{matrix} \right]=\left[ \begin{matrix} 5 & 6 \\ 1 & 8 \\ \end{matrix} \right]\Rightarrow \left[ \begin{matrix} 2 & 6 \\ 0 & 2x \\ \end{matrix} \right]+\left[ \begin{matrix} y & 0 \\ 1 & 2 \\ \end{matrix} \right]=\left[ \begin{matrix} 5 & 6 \\ 1 & 8 \\ \end{matrix} \right]\Rightarrow \left[ \begin{matrix} 2+y & 6+0 \\ 0+1 & 2x+2 \\ \end{matrix} \right]=\left[ \begin{matrix} 5 & 6 \\ 1 & 8 \\ \end{matrix} \right]$

Equating the corresponding elements, a11 and a22 we get

$2+y=5\Rightarrow \,\,\,y=3;\,\,\,2x+2=8\,\,\,\Rightarrow 2x=6\,\,\,\Rightarrow x=3;$

Hence x = 3 and y = 3 .

Illustration 3: Find the value of a, b, c and d, if $\left[ \begin{matrix} a-b & 2a+c \\ 2a-b & 3c+d \\ \end{matrix} \right]=\left[ \begin{matrix} -1 & 5 \\ 0 & 13 \\ \end{matrix} \right]$

Solution:

As the two matrices are equal, their corresponding elements are equal. Therefore, by equating the corresponding elements of given matrices we will obtain the value of a, b, c and d.

$\left[ \begin{matrix} a-b & 2a+c \\ 2a-b & 3c+d \\ \end{matrix} \right]=\left[ \begin{matrix} -1 & 5 \\ 0 & 13 \\ \end{matrix} \right]$

(given)

a – b = -1 … (i)

2a + c = 5 … (ii)

2a-b=0 … (iii)

3c + d = 13 … (iv)

Subtracting equation (i) from (iii), we have a = 1;

Putting the value of a in equation (i), we get b = 2

Putting the value of a in equation (ii), we have $2+c=5\Rightarrow c=3;$

Putting the value of c in equation (iv), we find $9+x=13\Rightarrow d=4$

Hence a=1,b=2,c=3,d=4.

 

Illustration 4: find x and y, if $2x+3y=\left[ \begin{matrix} 2 & 3 \\ 4 & 0 \\ \end{matrix} \right] and 3x+2y=\left[ \begin{matrix} 2 & -2 \\ -1 & 5 \\ \end{matrix} \right]$

Solution:

Solving the given equations simultaneously, we will obtain the values of x and y.

We have $2x+3y=\left[ \begin{matrix} 2 & 3 \\ 4 & 0 \\ \end{matrix} \right]$ … (i)

$3x+2y=\left[ \begin{matrix} 2 & -2 \\ -1 & 5 \\ \end{matrix} \right]$ … (ii)

Multiplying (i) by 3 and (ii) by 2, we get$6x+9y=\left[ \begin{matrix} 6 & 9 \\ 12 & 0 \\ \end{matrix} \right]$ … (iii)

$6x+4y=\left[ \begin{matrix} 4 & -4 \\ -2 & 10 \\ \end{matrix} \right]$ … (iv)

Subtracting (iv) from (iii), we get $5y=\left[ \begin{matrix} 6-4 & 9+4 \\ 12+2 & 0-10 \\ \end{matrix} \right]=\left[ \begin{matrix} 2 & 13 \\ 14 & -10 \\ \end{matrix} \right]$ $\Rightarrow y=\left[ \begin{matrix} \frac{2}{5} & \frac{13}{5} \\ \frac{14}{5} & \frac{-10}{5} \\ \end{matrix} \right]\Rightarrow y=\left[ \begin{matrix} \frac{2}{5} & \frac{13}{5} \\ \frac{14}{5} & -2 \\ \end{matrix} \right]$

Putting the value of y in (iii), we get $2x+3\left[ \begin{matrix} \frac{2}{5} & \frac{13}{5} \\ \frac{14}{5} & -2 \\ \end{matrix} \right]=\left[ \begin{matrix} 2 & 3 \\ 4 & 0 \\ \end{matrix} \right]$ $\Rightarrow 2x=\left[ \begin{matrix} 2 & 3 \\ 4 & 0 \\ \end{matrix} \right]-\left[ \begin{matrix} \frac{6}{5} & \frac{39}{5} \\ \frac{42}{5} & -6 \\ \end{matrix} \right]=\left[ \begin{matrix} 2-\frac{6}{5} & 3-\frac{39}{5} \\ 4-\frac{42}{5} & 0+6 \\ \end{matrix} \right]=\left[ \begin{matrix} \frac{4}{5} & -\frac{24}{5} \\ -\frac{22}{5} & 6 \\ \end{matrix} \right]\Rightarrow x=\left[ \begin{matrix} \frac{2}{5} & -\frac{12}{5} \\ -\frac{11}{5} & 3 \\ \end{matrix} \right]$

Hence $x=\left[ \begin{matrix} \frac{2}{5} & -\frac{12}{5} \\ -\frac{11}{5} & 3 \\ \end{matrix} \right] \;and\; y=\left[ \begin{matrix} \frac{2}{5} & \frac{13}{5} \\ \frac{14}{5} & -2 \\ \end{matrix} \right]$

ELEMENTARY ROW OPERATION OR GOUSS JORDON METHOD 

Elementary Row Transformations: ... Add one row to another row multiplied by a non zero number: Each element of a row is replaced by a number obtained by adding it to the scaled element of another row. It is symbolically represented as Ri → Ri + k Rj.

The Gauss Jordan Method is similar to Gaussian Elimination, except that the entries both above and below each pivot are targeted (zeroed out). After performing Gaussian Elimination on a matrix, the result is in row echelon form. After the Gauss-Jordan Method, the result is in reduced row echelon form.

What is the difference between Gauss elimination and Gauss Jordan method?

Gaussian Elimination helps to put a matrix in row echelon form, while Gauss-Jordan Elimination puts a matrix in reduced row echelon form. For small systems (or by hand), it is usually more convenient to use Gauss-Jordan elimination and explicitly solve for each variable represented in the matrix system

https://www.youtube.com/watch?v=BaHthPkpwoM

https://www.youtube.com/watch?v=q061E6QMCs8

https://www.youtube.com/watch?v=eYSASx8_nyg

https://www.youtube.com/watch?v=oPkTasoJngA

https://www.youtube.com/watch?v=AhUyh-2aPEc