Mathematics Revision Of Matrices For NDA

Matrix

In Mathematics, `"a matrix"` (plural matrices) is a rectangular array of numbers, symbols or expressions, arranged in rows and columns. The individuals in a matrix, are called its `"elements"` or `"entries."`

Generally, matrix is, written in following way

`A= [ (a_(11) , a_(12) , cdots , a_(1n)), (a_(21) , a_(22) , cdots , a_(2n)), (vdots , vdots , vdots, vdots), (a_(m1) , a_(m2) , cdots , a_(mn))] = |a_(ij)|_(m xx n)`

The order of a matrix `A` is `m xx n`, where `m` is the number of rows and `n` is the number of columns.

`"NOTE"` Here `m xx n` does not indicate multiplication

Types of Matrices

The matrices are of following types:

`"(i) Row Matrix :"` A matrix which have only one row and any number of columns, is called a row matrix.

`"(ii) Column Matrix :"` A matrix is said to be a column matrix, if it has only one column and any number of rows.

`"(iii) Null Matrix or Zero Matrix :"` A matrix of order `m xx n` whose all elements are zero, is called a null matrix of order `m xx n`. It is denoted by `O`.

`"(iv) Square Matrix :"` It is a matrix in which number of rows is equal to number of columns. Thus, ` m xx n` matrix is said to be a square matrix, if `m = n` and is known as a square matrix of order `n`.

`"(v) Diagonal Matrix :"` A square matrix is called a diagonal matrix, if all its non-diagonal elements are zero and diagonal elements are not all equal.

`"(vi) Scalar Matrix : "` A square matrix `A= |a_(ij)|` is said to be scalar matrix, if
(a) `a_(ij) = 0, AA i ne j`
(b) `a_(ij)= k, AA i = j`, where `k ne 0`
i.e. a diagonal matrix is said to be a scalar matrix, if the elements of principal diagonal are same.

`"(viii) Identity Matrix or Unit Matrix :"` A square matrix `A= |a_(ij)|` is said to be a unit or identity matrix, if
(a) `a_(ij) = 0, AA i ne j`
(b) `a_(ij) = 1, AA i= j`
i.e. a. diagonal matrix whose elements of principal diagonal are equal to `1` and all remaining elements are zero, is known as unit or identity matrix. It is denoted by `I`.

`"(viii) Upper Triangular Matrix :"` A square matrix `A =(a_(ij))_(n xx n)` if `a_(ij) =0` for `i > j` I.e. all elements below the leading diagonal are zero, is called upper triangular matrix.

`"(ix) Lower Triangular Matrix : "` A square matrix `A = (a_(ij) )_(m xx n)` if `a_(ij) = 0` for `i < j` i.e. all elements above the leading diagonal are zero, is called lower triangular matrix.

`"(x) Trace of a Matrix :"` The sum of diagonal elements of a square matrix A is called the trace of A and is denoted by `tr (A)`.

(a) `tr(lambda A) = lambda tr(A) `
(b) `tr(A) = tr(A')`
(c) `tr(AB) = tr(BA)`

`"(xi) Equal Matrices :"` Two matrices are said to be equal, if
(a) they have same number of rows and columns.
(b) The elements in the first matrix corresponding the elements of second matrix are equal.
e.g. Let `A= (a_(ij))_(m xx n)` and `B = (b_(ij) )_(m xx n)`

then, `A = B`, if `a_ij = b_(ij), AA i,j`

Addition of Matrices

Let `A= |a_(ij)|_(m xx n)` and `B = |b_(ij)|_(m xx n)` be two matrices of same order, then

` A+ B =|a_(ij)+b_(ij) |, = 1,2, ... ,m` and `j =1,2, ... ,n`

Properties of Addition of Matrices :

Let `A, B` and `C` be three matrices of same order, then the properties of addition of matrices are given below :

(i) Commutative `A + B = B + A`

(ii) Associative `(A + B)+ C =A+ (B +C)`

(iii) Additive Identity `A + O =A = O + A`

(iv) Additive Inverse `A+(- A)= O =(-A)+ A`

(v) Cancellation Laws :
(a) `A + B =A + C => B = C` [left cancellation law]
(b) `B +A= C +A=> B = C` [right cancellation law]

Subtraction of Matrices

Let `A` and `B` be two matrices of same order `m xx n`. then `A- B = [a_(ij)- b_(ij)] _(m xx n)`

Multiplication of a Matrix by a Scalar

Let `A =| a_(ij)|` be an `m xx n` matrix and `k` be any scalar. Then, the matrix obtained by multiplying each element of A by `k` is called the scalar multiple of `A` by `k`.

`"Properties of Multiplication of a Matrix by a Scalar"`

Let `A` and `B` he two matrices of same order, then the properties of multiplication of a matrix by scalar are
(i) `k(A+ B)=kA +kB`
(ii) `(k_1 + k_2 )A= k_1A + k_2A`

Multiplication of Matrices

Let `A= [a_(ij)]_(m xx n)` and `B = [b_(ij)]_(n xx p)` be two matrices such that number of columns of `A` is equal to the number of rows of `B`, then the product matrix is `C = [C_(ij)]` of order `m xx p`

where, `C_(ij) = sum_(k=1)^n a_(ik) b_(kj)`

`"Properties of Multiplication of Matrices:"`

Let `A, B` and `C` be three matrices of order `m xx n, n xx p` and `n xx k`, then the properties of multiplication of matrices are given below :

(i) Non-commutative `AB ne BA`

(ii) Associative `(AB)C = A(BC)`

(iii) Multiplicative Identity `IA = A =A I`

(iv) Multiplicative Distributive `A(B +C)= AB + AC`

Elementary Operation (Transformation) of a Matrix

Two matrices `A` and `B` are said to be equivalent, if one is obtained from the other by one or more elementary operations and we write `A ~ B`.

Following elementary operations are given below:

(i) Interchanging any two rows or columns is indicated by `R_i leftrightarrow R_j` or `C_i leftrightarrow C_j`

(ii) Multiplication of the elements of any row or column by a non-zero scalar quantity is indicated by

`R_i leftrightarrow k R_i`
or `C_i leftrightarrow k C_i`

(iii) Addition of constant multiple of the elements of any row or column to the corresponding element of any other row, is indicated by

`R_i -> R_i +k R_j`

or `C_i-> C_i + k C_j`

Special types of Matrices

`"1. Transpose of a Matrix :"`
Let `A` be `m xx n` matrix. Then, `n xx m` matrix obtained by interchanging the rows and columns of A is called the transpose of `A` and is denoted by `A'` or `A^c` or `A^T`.

`"Important Results :"`
If `A` and `B` are two matrices of order `m xx n`, then
`(A ± B)'= A' ± B'`

If `k` is a scalar, then `(k A)' = kA'`
`(A')'= A`
`(AB)' = B' A' `
`(A^n)' = (A')^n`

`"2. Idempotent Matrix :"`
A square matrix `A` is called an idempotent matrix, if it satisfies the relation `A^2 =A`. If `A` and `B` are idempotent matrices, then `A + B` is an idempotent iff `AB = BA`.


`"3. Nilpotent Matrix :"`
A square matrix `A` is called nilpotent matrix, if it satisfies the relation `A^k = 0` and `A^(k+1) ne O`.


`"4. Orthogonal Matrix :"`
A square matrix `A` is called an orthogonal matrix, if it satisfies the relation important Results `A A' = I`.

Important Results:
`=>` If `A` and `B` are orthogonal matrices, then `AB` is also an orthogonal matrix.
`=>` Every orthogonal matrix is invertible

`"5. Involutory Matrix :"`
A square matrix `A` is called an involutory matrix, if it satisfies the relation `A^2 = I`.

`"6. Symmetric Matrix : "`
A square matrix `A` is called symmetric matrix, if it satisfies the relation `A'= A`.

Important Results:
If `A` and `B` are symmetric matrices of the same order, then
(i) `AB` is symmetric if and only if `AB = BA`.
(ii) `A ± B, AB + BA` are also symmetric matrices.

If `A` is symmetric matrix, then `A^(-1)` will also be symmetric matrix.

`"7. Skew symmetric Matrix :"`
A square matrix `A` is called skew-symmetric matrix, if it satisfies the relation `A'=-A`. Every square matrix can be uniquely expressed as the sum of symmetric and skew-·symmetric matrix.

i.e. `A= 1/2 (A+ A') + 1/2 (A- A')`

Adjoint of a Matrix

Let `A =[a_(ij)]_(m xx n)` be a square matrix of order `n` and `C_(ij)` be the cofactor of `a_(ij)` in the determinant `|A|`. Then, the adjoint of `A` is defined as the transpose of the cofactor matrix and is denoted by `adj (A)`.

Properties of Adjoint of a Matrix :

Let `A` be a matrix of order `n`, then

(i) `(adjA)A=A(adjA)=|A| * I_n`
(ii) `| adj A |= | A|^(n- 1)` , if `|A | ne 0`
(iii) `adj (AB) = (adj B) (adj A)`
(iv) If `|A| =0`, then `(adjA) A= A (adj A)= O`
(v) `adj(A^T) = (adj A )^T`
(vi) `|adj (adj A) |= |A|^((n-1)^2)`
(vii) Adjoint of a diagonal matrix is a diagonal matrix.
(viii) `adj (adj A)= |A|^(n-2) * A`

Singular and Non-singular Matrix

If `A` is a square matrix such that ` |A | = 0`, then matrix `A` is singular. However, if `|A| ne 0`, then the matrix `A` is called non-singular.

Inverse of a Matrix

Let `A` be a ,non-singular (where, `|A| ne 0`) square matrix. Then a square matrix `B` such that `AB = BA = I` is called
inverse of `A` i.e. `A^(-1) = 1/(|A|) adj (A)`

Properties of Inverse of a Matrix :

Let `A` and `B` be square matrices of same order, then
(i) A square matrix is invertible if and only if it is non-singuIar.
(ii) `(A ')^(-1)= (A^(-1))' `
(iii) ` (AB)^(-1) = (B^(-1)A^(-1))`
(iv) `|A^(-1)|= |A|^(-1)`

Solution of a System of Linear Equations

Let system of linear equations in three variables be

`a_1x + b_1y + c_1z = d1, a_2x + b_2y + c_2z = d_2`

and `a_3x + b_3 y + c_3 z = d_3`

It can be written in the matrix form as

`[(a_1 , b_1, c_1),(a_2, b_2, c_2),(a_3 , b_3 , c_3)] [ (x),(y),(z)]=[(d_1),(d_2),(d_3)]=> AX = B => X= A^(-1) B`

 
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