Properties of various types of matrices:
Given a square matrix A = [a_{ij}]_{n×n},
1) For upper triangular matrix, a_{ij} = 0, ∀ i > j
2) For lower triangular matrix, a_{ij }= 0, ∀ i < j
3) Diagonal matrix is both upper and lower triangular
4) A triangular matrix A = [a_{ij}]_{n×n} is called strictly triangular if a_{ii} = 0 for ∀
If A = [a_{ij}]_{m×n} and transpose of A i.e. A' = [b_{ij}]_{n×m} then b_{ij }=a_{ji}, ∀i, j.
Properties of Transpose:
1) (A')' = A
2) (A + B)' = A' + B', A and B being conformable matrices
3) (αA)' = αA', α being scalar
4) (AB)' = B'A', A and B being conformable for multiplication
Properties of Transpose conjugate:
1) (A^{θ})^{θ} = A
2) (A + B)^{θ }= A^{θ }+ B^{θ}
3) (kA)θ = A^{θ}, k being any number
4) (AB)θ = B^{θ}A^{θ}
Some chief properties of matrices:
1) Only matrices of the same order can be added or subtracted.
2) Addition of matrices is commutative as well as associative.
3) Cancellation laws hold well in case of addition.
4) The equation A + X = 0 has a unique solution in the set of all m × n matrices.
5) All the laws of ordinary algebra hold for the addition or subtraction of matrices and their multiplication by scalar.
Matrix Multiplication:
1) Matrix multiplication may or may not be commutative. i.e., AB may or may not be equal to BA
2) If AB = BA, then matrices A and B are called Commutative Matrices.
3) If AB = BA, then matrices A and B are called AntiCommutative Matrices.
4) Matrix multiplication is Associative
5) Matrix multiplication is Distributive over Matrix Addition.
6) Cancellation Laws need not hold goodin case of matrix multiplication i.e., if AB = AC then B may or may not be equal to Ceven if A ≠ 0.
7) AB = 0 i.e., Null Matrix, does not necessarily imply that either A or B is a null matrix.
A square matrix A = [a_{ij}] is said to be symmetric when aij = aji for all i and j.
If a_{ij }= a_{ji} for all i and j and all the leading diagonal elements are zero, then the matrix is called a skew symmetric matrix.
A square matrix A = [a_{ij}] is said to be Hermitian matrix if A^{θ} = A.
1) Every diagonal element of a Hermitian Matrix is real.
2) A Hermitian matrix over the set of real numbers is actually a real symmetric matrix.
A square matrix, A = [a_{ij}] is said to be a skewHermitian matrix if A^{θ }= A.
1) If A is a skewHermitian matrix then the diagonal elements must be either purely imaginary or zero.
2) A skewHermitian Matrix over the set of real numbers is actually a real skew symmetric matrix.
Any square matrix A of order n is said to be orthogonal if AA' = A'A = I_{n}.
A matrix such that A_{2} = I is called involuntary matrix.
Let A be a square matrix of order n. Then A(adj A) = A I_{n} = (adj A)A.
The adjoint of a square matrix of order 2 can be easily obtained by interchanging the diagonal elements and changing the signs of offdiagonal (left hand side lower corner to right hand side upper corner) elements.
A nonsingular square matrix of order n is invertible if there exists a square matrix B of the same order such that AB = I_{n} = BA.
The inverse of A is given by A^{1} = 1/A.adj A.
Properties of Inverse of a matrix:
1) Every invertible matrix possesses a unique inverse.
2) If A and B are invertible matrices of the same order, then AB is invertible and (AB)^{1} = B^{1}A^{1}. This is also termed as the reversal law.
3) In general,if A,B,C,...are invertible matrices then (ABC....)^{1} =..... C^{1} B^{1 }A^{1}.
4) If A is an invertible square matrix, then AT is also invertible and (AT)^{1} = (A^{1})^{T}.
The following three operations can be applied on rows or columns of a matrix:
1) Interchange of any two rows (columns)
2) Multiplying all elements of a row (column) of a matrix by a nonzero scalar. If the elements of i^{th} row (column) are multiplied by nonzero scalar k, it will be denoted by R_{l}→R_{i }(k) [C_{i}→C_{i} (k)] or Rl→kR_{i} [C_{i}→kC_{i}].
3) Adding to the elements of a row (column), the corresponding elements of any other row (column) multiplied by any scalar k.
A number ‘r’ is called the rank of a matrix if:
1) Every square sub matrix of order (r +1) or more is singular.
2) There exists at least one square sub matrix of order r which is nonsingular.
It also equals the number of nonzero rows in the row echelon form of the matrix.
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