Basis of Matrix Study Notes for Mechanical and Civil Engineering

By Akhil Gupta|Updated : November 5th, 2020



A system of m×n numbers arranged in a rectangular formation along m rows and n columns and bounded by the brackets [ ] is called an m by n matrix; which is written as m × n matrix. A matrix is also denoted by a single capital letter A.

A = [aij], i to m, j = 1 to n.



1.2 Row Matrix:

A matrix with only one row is called a row matrix.

E.g. = [1 2 4]1x3

1.3 Column Matrix:

A matrix with only one column is called a column matrix.


1.4 Square Matrix:

A square matrix has an equal number of rows and columns. (i.e.) m = n.


1.5 Diagonal Matrix:

A square matrix is called a diagonal matrix if each of its non-diagonal elements are zero (i.e.) A square matrix A is diagonal if aij ≠ 0 for i = j and

aij = 0 for i ≠ j.


1.5.1. Properties of diagonal Matrix:

(a)  diag [x, y, z] + diag [p, q, r] = diag [x + p, y + q, z + r]

(b)  diag [x, y, z] × diag [p, q, r] = diag [xp, yq, zr]

(c)  (diag [x, y, z])–1 = diag [1/x, 1/y/ 1/z]

(d)  (diag[x, y, z])T = diag[x, y, z]

(e)  diag [x, y, z]n = diag[xn, yn, zn]

(f)  Eigen values of diag [x, y, z] = x, y and z.

(g)  Determinant of diag [x, y, z] = | diag[x, y, z]| = xyz

1.6 Scalar Matrix:

A diagonal matrix whose diagonal elements are equal is called a scalar matrix.


1.7 Upper Triangular Matrix:

An upper triangular matrix is one whose entry values below the leading diagonal are zero.


1.8 Lower Triangular Matrix:

A lower triangular matrix is one whose entry values above the leading diagonal are zeros.


1.9 Equality of Matrices:

Two matrices are equal if they are of the same dimension and their corresponding entries are equal.

Eg: A = B

Where image008

1.10 Identity Matrix:

An identity matrix is one whose leading diagonal entries are ones all the other entries are zero.


1.10.1. Properties of Identity Matrix:

(a) I is identity element for multiplication, so it is called multiplicative identity

(b) AI = IA = A

(c) In = I

(d) I–1 = I

(e) |I| = 1

1.11 Transpose of a Matrix:

A matrix obtained from any given matrix A, by interchanging its rows and columns is called the transpose of A an dis usually denoted by A' (or) AT.

Thus if A = [aij], then A' = [bij] where bij = aij.


1.11.1 Properties of Transpose of a Matrix:

If AT and BT be transpose of A and B respectively then,

  1. (AT)T = A
  2. (A + B)T = AT + BT
  3. (kA)T = kAT, k being any real number.
  4. (AB)T= BTAT
  5. (ABC)T = CTBTAT

1.12 Conjugate of a Matrix:

The matrix obtained from given matrix A, on replacing its elements by the corresponding conjugate complex numbers is called the conjugate of A and denoted by A. Thus if A = [aij], then A = [bij] where bij = aij is called the conjugate of A.

Eg: image011

1.13 Conjugate Transpose of A Matrix:

The conjugate of the transpose of a matrix A is called its conjugate transpose and denoted by AT. Thus AT.


1.14 Symmetric Matrices:

A square matrix A = [aij] is said to be symmetric if aij = aji ∀i, j.

(i.e.) the (i,j)th element is same as the (j, i)th element.


Therefore A = A'.


  1. If A and B and symmetric, then:

(a) A + B and A – B are also symmetric

(b) AB, BA may or may not be symmetric.

(c) Ak is symmetric when k is set of any natural number.

(d) AB + BA is symmetric.

(e) AB – BA is skew symmetric.

(f) A2, B2, A2 ± B2 are symmetric.

(g) KA is symmetric where k is any scalar quantity.

1.15 Skew Symmetric Matrices:

A square matrix A = [aij] is said to be skew symmetric of aij = -aji.

(i.e.) the (i, j)th element is the negative of its (j, i)th element.


Therefore A = -A'.

1.16 Hermitian Matrices:

A square matrix A = [aij] is said to be Hermitian if aij = aji.

(i.e.) the (i, j)th elements is the conjugate complex of the (j, i)th elements.

(i.e.) for a Hermitian matrix A, we have A = A'.

1.17 Skew-Hermitian Matrices:

A square matrix A = [aij] is said to be Skew-Hermitian if aij = -aji.

(i.e.) the (i, j)th elements is the negative conjugate complex of the (j, i)th elements.


1.18 Sub Matrix:

The matrix obtained on deleting any number of rows and columns of the given matrix A is called the sub-matrix of A.


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