An m × n matrix is usually written as:
In brief, the above matrix is represented by A = [a_{ij}]_{ mxn}. The numbers a_{11}, a_{12}, ….. etc., are known as the elements of the matrix A, where a_{ij} belongs to the i^{th} row and j^{th} column and is called the (i, j)^{th} element of the matrix A = [a_{ij}].
If A and B are square matrices of order n, and I_{n} is a corresponding unit matrix, then
(a) A(adj.A) =  A  I_{n} = (adj A) A
(b)  adj A  =  A n^{1} (Thus A (adj A) is always a scalar matrix)
(c) adj (adj.A) =  A ^{n2} A
(d)
(f) adj (A^{m}) = (adj A)^{m},
(g) adj(kA) = k^{n1 }(adj.A), KϵR
(h) adj (I_{n}) = I_{n}
(i) adj 0 = 0
(j) A is symmetric ⇒adj A is also symmetric
(k) A is diagonal ⇒adj A is also diagonal
(l) A is triangular ⇒adj A is also triangular
(m) A is singular ⇒ adj A  = 0
(i) Symmetric matrix: A square matrix A = [a_{ij}] is called a symmetric matrix if a_{ij} = a_{ji}, for all i, j.
(ii) Skewsymmetric matrix: when a_{ij} = – a_{ji}
(iii) Hermitian and skew – Hermitian matrix:
(Hermitian matrix)(A^{θ} represents conjugate transpose)
(skewHermitian matrix)
(iv) Orthogonal matrix: if AA^{T} = I_{n} = A^{T}A
(v) Idempotent matrix: if A^{2} = A
(vi) Involuntary matrix: if A^{2} = I or A^{1} = A
(vii) Nilpotent matrix: A square matrix A is nilpotent; if A^{p }= 0, p is an integer.
The trace of a square matrix is the sum of the elements on the main diagonal.
(i) tr(λA_ = λ tr(A)
(ii) tr(A + B) = tr(A) + tr(B)
(iii) tr(AB) = tr(BA)
Properties of Matrix Multiplication
(i) AB ≠ BA
(ii) (AB)C = A(BC)
(iii) A.(B + C) = A.B + A.C
A matrix which has m rows and n columns is called a matrix of order m x n.
For example, the order of
matrix is 2 x 3.
Note: (a) The matrix is just an arrangement of certain quantities.
(b) The elements of a matrix may be real or complex numbers. If all the elements of a matrix are real, then the matrix is called a real matrix.
(c) An m x n matrix has m.n elements.
Illustration 1: Construct a 3×4 matrix A = [a_{ij}], whose elements are given by a_{ij} = 2i + 3j.
Sol: In this problem, I and j are the number of rows and columns, respectively. By substituting the respective values of rows and columns in a_{ij} = 2i + 3j, we can construct the required matrix.
Given a_{ij} = 2i + 3j
so a_{11} = 2+3 = 5, a_{12} = 2+6 = 8
so a_{11} = 2+3 = 5, a_{12} = 2+6 = 8
Similarly, a_{13} = 11, a_{14}=14, a_{21} = 7, a_{22}=10, a_{23}=13, a_{24}=16,a_{31}=9, a_{32}=12, a_{33}=15, a_{34}=18
Illustration 2: Construct a 3 x 4 matrix, whose elements are given by: aij =
Sol: The method for solving this problem is the same as in the above problem.
Since
Hence, the required matrix is given by
Let A = [a_{ij}]_{nxn} and B = [b_{ij}]_{nxn} and λ be a scalar,
(i) tr(λA) = λ tr(A) (ii) tr(A + B) = tr(A) + tr(B) (iii) tr(AB) = tr(BA)
The matrix obtained from a given matrix A by changing its rows into columns or columns into rows is called the transpose of matrix A and is denoted by AT or A’. From the definition, it is obvious that if the order of A is m x n, then the order of AT becomes n x m; For example, transpose of a matrix.
(i) (A^{T})^{T}= A
(ii) (A + B)^{T} = A^{T}+ B^{T}
(iii) (AB)^{T} = B^{T}A^{T}
(iv) (kA)^{T} = k(A)^{T}
(v) (A_{1}A_{2}A_{3} ……A_{n1}A_{n})^{T} =
(vi) I^{T} = I (vii) tr(A) = tr(A^{T})
Illustration 3: If . then prove that (AB)T = BTAT.
Solution: By obtaining the transpose of AB, i.e., (AB)^{T} and multiplying B^{T} and A^{T}, we can easily get the result.
Here, AB =
Illustration 4: If Then what is (B’)’A’ equal to?
Sol: In this problem, we use the properties of the transpose of a matrix to get the required result.
We have =
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1. What is a matrix? 
2. How do you add two matrices together? 
3. What is the determinant of a matrix? 
4. How do you multiply two matrices together? 
5. What is the inverse of a matrix? 
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