Basis of Matrix Study Notes for Mechanical and Civil Engineering

1. MATRIX

1.1.Definition:

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.

or

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 

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) A^T.

Thus if A = [a_ij], then A' = [b_ij] where b_ij = a_ij.

1.11.1 Properties of Transpose of a Matrix:

If A^T and B^T be transpose of A and B respectively then,

1. (A^T)^T = A
2. (A + B)^T = A^T + B^T
3. (kA)^T = kA^T, k being any real number.
4. (AB)^T= B^TA^T
5. (ABC)^T = C^TB^TA^T

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 = [a_ij], then A = [b_ij] where b_ij = a_ij is called the conjugate of A.

Eg: 

1.13 Conjugate Transpose of A Matrix:

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

1.14 Symmetric Matrices:

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

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

Therefore A = A'.

Note.1: 

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) A^k 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 = [a_ij] is said to be skew symmetric of a_ij = -a_ji.

(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 = [a_ij] is said to be Hermitian if a_ij = a_ji.

(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 = [a_ij] is said to be Skew-Hermitian if a_ij = -a_ji.

(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.

Online Classroom Program for ME (24+ structured Courses & 161+ Mock Tests)

Online Classroom Program for CE (20+ Structured Courses & 180+ Mock Tests)

Thanks

The Most Comprehensive Exam Prep App.