Matrices Formulas

Order of Matrix

A matrix which has m rows and n columns is called a matrix of order m × n.

Types of Matrices

(i) Row Matrix:
If in a matrix, there is only one row, then it is called a Row Matrix.
Thus A = [a_ij]_m×n is a Row Matrix if m = 1.

(ii) Column Matrix:
If in a matrix, there is only one column, then it is called a Column Matrix.
Thus A = [a_ij]_m×n is a Column Matrix if n = 1.

(iii) Square Matrix:
If number of rows and number of columns in a matrix are equal, then it is called a Square Matrix.
Thus A = [a_ij]_m×n is a Square Matrix if m = n

(iv) Trace of a Matrix:
The sum of diagonal elements of a Square Matrix A is called the Trace of Matrix A which is denoted by tr A = \(\sum_{i=1}^{n} a_{i j}\) = a₁₁ + a₂₂ + ……. + a_nn

(v) Singleton Matrix:
If in a matrix there is only one element then it is called Singleton Matrix.

(vi) Null or Zero Matrix:
If in a matrix all the elements are zero then it is called a Zero Matrix and it is generally denoted by 0. Thus A = [a_ij]_m×n is a Zero Matrix if a_ij = 0 for all i and j.

(vii) Diagonal Matrix:
If all elements except the principal diagonal in a square matrix are zero, it is called a Diagonal Matrix. Thus a square matrix A = [a_ij] is a diagonal matrix if a_ij = 0 when i ≠ j.

(viii) Scalar Matrix:
If all the elements of the diagonal of a diagonal matrix are equal, it is called a Scalar Matrix.

(ix) Unit Matrix:
If all elements of principal diagonal in a diagonal matrix are 1, then it is called Unit Matrix. A Unit Matrix of order n is denoted by I_n.

(x) Triangular Matrix:
A square matrix [a_ij] is said to be Triangular Matrix if each element above or below the principal diagonal is zero it is of two types-

• Upper Triangular Matrix: A square matrix [a_ij] is called the Upper Triangular Matrix, if a_ij = 0 when i > j.
• Lower Triangular Matrix: A square matrix [a_n] is called the Lower Triangular Matrix, if a_ij = 0 when i < j.

(xi) Singular Matrix:
Matrix A is said to be Singular Matrix if its determinant |A| = 0, otherwise non-singular matrix i.e.
If det |A| = 0 ⇒ Singular and det |A| ≠ 0 ⇒ non-singular

Addition and Subtraction of Matrices

If A = [a_ij]_m×n and B = [b_ij]_m×n are two matrices of the same order then their sum A + B is a matrix whose each element is the sum of corresponding element.

Scalar Multiplication of Matrices

Let A = [a_ij]_m×n be a matrix and k be a number then the matrix which is obtained by multiplying every element of A by k is called scalar multiplication of A by k and it is denoted by kA thus if A = [a_ij]_m×n then kA = Ak = [ka_ij]_m×n
Properties of Scalar Multiplication
If A, B are Matrices of the same order and λ, μ are any two scalars then-

• λ(A + B) = λA + λB
• (λ + μ) A = λA + μ A
• λ(μA) = (λμA) = μ (λA)
• (- λA) = – (λA) = λ (-A)
• tr (kA) = k tr(A)

Multiplication of Matrices

If A and B be any two matrices, then their product AB will be defined only when number of column in A is equal to the number of rows in B. If A = [a_ij]_m×n and B = [b_ij]_m×n then their product AB = C = [c_ij]_m×n will be matrix of order m x p, where (AB)_ij = C_ij = \(\sum_{r=1}^{n} \alpha_{i r} b_{r j}\)
Properties of Matrix Multiplication
If A, B and C are three matrices such that their product is defined, then

(i) AB ≠ BA	(Generally not commutative)
(ii) (AB) C = A (BC)	(Associative Law)
(iii) IA = A = AI	(I is identity matrix for matrix multiplication)
(iv) A (B + C) = AB + AC	(Distributive Law)
(v) If AB = AC ⇏ B = C	(Cancellation nLaw is not applicable)
(vi) If AB = 0	It does not mean that A = 0 or B = 0, again product of two non- zero matrix may be zero matrix.
(vii) tr (AB) = tr (BA)

Positive Integral powers of a Matrix

The positive integral powers of a matrix A are defined only when A is a square matrix.
Also then A² = A.A, A³ = A.A.A = A²A
Also for any positive integers m, n

• A^mAⁿ = A^m+n
• (A^m)ⁿ = A^mn = (Aⁿ)^m
• Iⁿ = I,I^m = I
• A° = I_n where A is a square matrices of order n.

Transpose of Matrix

If order of A is m × n, then the order of A^T is n × m.
Properties of Transpose

• (A^T)^T = A
• (A ± B)^T = A^T ± B^T
• (AB)^T = B^TA^T
• (kA)^T = k(A)^T
• (A₁A₂A₃ ……. A_n-1A_n)^T = A_n^T A_n-1^T …….. A₃^TA₂^TA₁^T
• I^T = I
• tr (A) = tr (A^T)

Symmetric Matrix

A square matrix A = [a_ij] is called Symmetric Matrix if a_ij = a_ji for all i, j or A^T = A

Skew-Symmetric Matrix

A square matrix A =[a_ij] is called skew – symmetric matrix if
a_ij = – a_ij for all i, j.
Every square matrix A can uniquely be expressed as sum of a symmetric and Skew Symmetric Matrix i.e.
A = \(\left[\frac{1}{2}\left(\mathrm{A}+\mathrm{A}^{\mathrm{T}}\right)\right]+\left[\frac{1}{2}\left(\mathrm{A}-\mathrm{A}^{\mathrm{T}}\right)\right]\)

Adjoint of a Matrix

If A = [a_ij] be a square matrix and F_ij be the cofactor of a_ij in [A|, then Adj. A = [F_ij]^T
Properties of adjoint A

• If A be n rowed square matrix then (adj A) A = A(Adj A) = |A|. I_n
• If A is singular matrix then (adj A) A = A(adj A) = 0
• If A is n rowed square non singular matrix then |adj A| = |A|^n-1
• Adj (AB) = (Adj B). (Adj A)
• (Adj A)’= Adj (A’)
• (adj A)’ = |A|^n-2 A where A is a non singular matrix.
• |Adj-(adj A)|= |A|^(n-2)²

Inverse of a Matrix

If A & B are two matrices such that AB = I = BA then B is called the inverse of A and it is denoted by A^-1, thus A^-1 = B <=> AB = I = BA To find inverse matrix of a given matrix A we use following formula
A^-1 = \(\frac{\text { Adj } A}{|\mathrm{A}|}\), thus A^-1 exists ⇔ |A| ≠ 0

Some special cases of Matrices

(i) Orthogonal Matrix:
A square matrix A is called Orthogonal if AA^T = I = A^T A

(ii) Idempotent Matrix:
A square matrix A is called an Idempotent Matrix if A² = A

(iii) Involutory Matrix:
A square matrix A is called an involutory Matrix if A² = I or A^-1 = A

(iv) Nilpotent Matrix:
A square matrix A is called a Nilpotent Matrix if there exist a p ∈ N such that A^p = 0

(v) Hermitian Matrix:
A square matrix is Hermitian Matrix if A^θ = A i.e.a_ij \(=\bar{a}_{j i}\) ∀ i, j

(vi) Skew Hermitian Matrix:
A square matrix A is Skew-Hermitian ifA = -A^θ i.e. aij = –\(\bar{a}_{j i}\) ∀ i, j

(vii) Period of a Matrix:
If for any matrix A
A^k+1 = A then k is called Period of Matrix (where k is a least positive integer)

(viii) Differentiation of a Matrix
If A = \(A=\left[\begin{array}{ll}f(x) & g(x) \\h(x) & \ell(x)\end{array}\right]\) then \(\frac{\mathrm{d} \mathrm{A}}{\mathrm{dx}}=\left[\begin{array}{ll}\mathrm{f}^{\prime}(\mathrm{x}) & \mathrm{g}^{\prime}(\mathrm{x}) \\\mathrm{h}^{\prime}(\mathrm{x}) & \ell^{\prime}(\mathrm{x})\end{array}\right]\) is a differentiation of Matrix A

(ix) Rank of a Matrix:
A number r is said to be the rank of a m × n matrix A if

• Every square sub matrix of order (r + 1) or more is singular and
• There exists at least one square sub matrix of order r which is non- singular.

Thus, the rank of matrix is the order of the highest order nonsingular sub matrix.
eg. The Rank of Matrix A = \(\left[\begin{array}{lll}1 & 2 & 3 \\4 & 5 & 6 \\3 & 4 & 5\end{array}\right]\)
We have |A| = 0, therefore r(A) is less than 3, we observe that \(\left[\begin{array}{ll}5 & 6 \\4 & 5\end{array}\right]\) is a non-singular square sub matrix of order 2. Hence, r(A) = 2.

(x) Rank of zero matrix is zero also rank of non singular matrix of order “n” is “n”.

Type of Equations

(a) If |A| ≠ 0 system of equation is always consistent and in case of homogeneous equation, system has only “trivial solution and in case of non homogeneous equation, unique solution can be given by using x = A^-1B

(b) If |A| = 0, In case of homogeneous equation system has nontrivial (infinite many) solutions and in case of non homogeneous system of equation. If (adj A) B ≠ 0 then system is consistent otherwise i.e. (adj A) B = 0 system of equation is consistent and has infinite no. of solutions.
Note:
If No. of equation < No. of unknowns, then it has non trivial solutions.

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