Importance of Linear Algebra in Engineering Mathematics
linear algebra provides a powerful framework for solving systems of linear equations. Engineers encounter numerous scenarios where multiple equations must be solved simultaneously to determine unknown variables. Through techniques like matrix algebra, Gaussian elimination, and matrix inverses, linear algebra equips engineers with the ability to efficiently tackle such systems and find solutions that satisfy the given conditions.
One of the fundamental concepts in linear algebra is the vector, which represents both magnitude and direction. Vectors find extensive use in engineering, aiding in the representation of forces, velocities, displacements, and electrical currents, among other quantities. Understanding vector operations, such as addition, subtraction, and scalar multiplication, enables engineers to manipulate and analyze these quantities effectively.
Linear transformations form another crucial aspect of linear algebra. These transformations capture the fundamental behavior of systems by mapping vectors from one space to another while preserving their linear relationships. Engineers employ linear transformations to model and analyze dynamic systems, such as electrical circuits, control systems, and mechanical structures. Linear algebra is an indispensable tool in engineering, enabling professionals to analyze, model, and solve a broad spectrum of complex problems. Its concepts and techniques find applications across various disciplines, allowing engineers to develop innovative solutions that drive advancements in technology and improve our daily lives. By harnessing the power of linear algebra, engineers unlock the potential to transform theoretical concepts into practical, tangible realities.
What is Matrix?
Matrix is one of the important topics of the engineering mathematics linear algebra. It is defined as a system of mn numbers arranged in a rectangular formation along m rows and n columns and bounded by the brackets [ ] and is called an m by n matrix, written as an m × n matrix. A matrix is also represented by a single capital letter A.
Thus, A is a matrix of order mn.
It has m rows and n columns. The element of the Matrix is Each of the mn numbers.
Special Matrices and their Properties
Transpose of a Matrix
The Matrix acquired from any given matrix A, by exchange rows and columns, is called the transpose of A and is denoted by AT or A’. The transpose of a matrix matched with itself, i.e. (A’)’ = A.
Row and Column Matrix
- A matrix having a single row is called a row matrix.
- A matrix having a single column is called a column matrix.
- Row and column matrices are sometimes called row vectors and column vectors.
Square Matrix
- An m × n matrix for which the number of rows equals the number of columns, i.e. m = n, is called a square matrix.
- It is also called an n-rowed square matrix.
- The element aij such that i = j, i.e. a11, a22… are called DIAGONAL ELEMENTS and the line along which they line is called the Principle Diagonal of the Matrix.
- Elements other than principal diagonal elements are called off-diagonal elements, i.e. aij such that I ≠ j.
Diagonal Matrix
A square matrix in which all off-diagonal elements are zero is called a diagonal matrix. The diagonal elements may be zero or may not be zero.
Scalar Matrix
A scalar matrix is a diagonal matrix with all diagonal elements belonging equal.
A square matrix, each of whose diagonal elements is 1 and each of whose non-diagonal elements are zero, is called a unit matrix or an identity matrix which is denoted by I.
- The identity matrix is always square.
- Thus, a square matrix A = [aij] is a unit matrix if aij = 1 when i = j and aij = 0 when i ≠ j.
Null Matrix
- The m × n matrix with zero elements is called a null matrix. The null Matrix is denoted by O.
- Null Matrix need not be square.
Upper Triangular Matrix
- An upper triangular matrix is a square matrix whose lower off-diagonal elements are zero i.e. aij = 0 whenever i > j.
- U denotes it.
- The diagonal and upper off-diagonal elements may or may not be zero.
Lower Triangular Matrix
- A lower triangular matrix is a square matrix whose upper off-diagonal triangular elements are zero, i.e., aij = 0 whenever i < j.
- The diagonal and lower off-diagonal elements may or may not be zero. L denotes it.
Idempotent Matrix
A matrix A is called idempotent if A2 = A.
Involutory Matrix
A matrix A is called involutory if A2 = I.
Nilpotent Matrix
A matrix A is said to be nilpotent of class m or index m if Am = 0 and Am – 1 ≠ 0, i.e., m is the smallest index which makes Am = 0
Singular Matrix
A matrix will be a singular matrix if its determinant equals zero.
Periodic Matrix
A square matrix A is periodic if Ak + 1 = A where k is the least positive integer and is called the period of A.
Classification of Real Matrices
Real matrices can be classified into three types of relationships between AT and A.
Symmetric Matrix
- A square matrix A = [aij] is said to be symmetric if its (i, j)th elements are the same as its (j, i)th element i.e. aij = aij for all i and j.
- In a symmetric matrix: AT = A
Properties of symmetric matrices
For any Square matrix A,
If A and B and symmetric, then:
(a) A + B and A – B are also symmetric
(b) AB, BA may be symmetric or may not be symmetric.
(c) Ak is symmetric when a set of k is 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 a scalar quantity.
Skew – Symmetric Matrix
- A square matrix A = [aij] is said to be skew-symmetric if (i, j)th elements of A are the negative of the (j, i)th elements of A if aij = –aij ∀ i, j.
- In a skew-symmetric matrix, AT = –A.
- A skew-symmetric matrix must have all 0’s in the diagonal.
If A and B and symmetric, then:
(a) A ± B is skew-symmetric.
(b) AB and BA are not skew-symmetric.
(c) A2, B2, A2 ± B2 are symmetric.
(d) A2, A4, and A6 are symmetric.
(e) A3, A5, and A7 are skew-symmetric.
(f) kA is skew-symmetric, where k is any scalar number.
Orthogonal Matrices
A square matrix A is said to be orthogonal if: AT = A–1 ⇒ AAT = AA–1 = 1. Thus, A will be an orthogonal matrix if:
AAT = I = ATA.
Trace of a Matrix
Let A be a square matrix of order n. Then, the Sum of elements lying in the principal diagonal is called the trace of A, denoted by Tr(A).
Thus, if A = [aij]n×n
Properties of the trace of the Matrix
(a). tr (λA) = λ tr(A)
(b). tr (A +B) = tr (A) + tr (B)
(c). tr (AB) = tr (BA)
The Inverse of a matrix
If A be any matrix, then a matrix B if it exists, such that:
AB = BA = I
Then, B is called the Inverse of A, denoted by A-1 so that AA-1= I.
Properties of Inverse
(a). AA–1 = A–1 A = I
(b). A and B are inverses of each other if AB = BA = I
(c). (AB)–1 = B–1 A–1
(d). (ABC)–1 = C–1 B–1 A–1
(e). If A is an n × n non-singular matrix, then (A)–1 = (A–1)’.
The Rank of Matrix
The rank of a matrix is defined as the order of the highest non-zero minor of matrix A. It is denoted by the notation ρ(A). A matrix is said to be of rank r when:
(i) it has at least one non-zero minor of order r, and
(ii) every minor of order higher than r vanishes.
Properties
(a). Rank of A and its transpose is the same i.e.
(b). Rank of a null matrix is zero.
(c). Rank of a non-singular square matrix of order r is r.
(d). If a matrix has a non-zero minor of order r, its rank is r, and if all minors of a matrix of order r + 1 are zero, its rank is r.
(e). Rank of a matrix is the same as the number of linearly independent row vectors in the matrix X and the number of linearly independent column vectors in the matrix.
(f). For any matrix A, rank (A) min(m,n) i.e. maximum rank of Am×n = min (m, n).
(g). If Rank (AB) Rank A and Rank (AB) Rank B:
so, Rank (AB) ≤ min (Rank A, Rank B)
(h). Rank (AT) = Rank (A)
(i). Rank of a matrix is the number of non-zero rows in its echelon form.
(j). Elementary transformations do not alter the rank of a matrix.
(k). Only the null matrix can have a rank of zero. All other matrices have the rank of at least one.
(l). Similar matrices have the same rank.
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