Inverse of Matrix
Inverse of Matrix
The inverse of a matrix A, exists if A is non-singular (i.e., ⏐A⏐ ≠ 0) and is given by the formula
A–1 = Adj (A) / |A|
The inverse of a matrix is always unique.
Adjoint of a Square Matrix
Let A=[aij] be any n × n matrix. The transpose B of the matrix B = [Aij]n x n, where Aij denotes the cofactor of element aij is called the adjoint of matrix A and is denoted by the symbol Adj A.
∴ Adj (A) = [cof (A)]T
Properties of Adjoint:
If A be any n-rowed square matrix, then (Adj A) A = A (Adj A) = ⏐A⏐ In
where In is the n × n identity matrix.
Rank of A Matrix
Rank is defined for any matrix Am × n (need not be square)
Some important concepts:
1. Submatrix of a Matrix: Suppose A is any matrix of the type m × n. Then a matrix obtained by leaving some rows and some columns from A is called a submatrix of A.
2. Rank of a Matrix: A number r is said to be the rank of a matrix A, if it possesses the following properties:
(a) There is at least one square sub-matrix of A of order r whose determinant is not equal to zero.
(b) If the matrix A contains any square sub-matrix of order (r + 1) and above, then the determinant of such a matrix should be zero.
Put together, property (a) and (b) give the definition of the rank of a matrix as the “order of the largest non-zero minor.”
Dear Aspirants,
Your preparation for GATE, ESE, PSUs, and AE/JE is now smarter than ever — thanks to the MADE EASY YouTube channel.
This is not just a channel, but a complete strategy for success, where you get toppers strategies, PYQ–GTQ discussions, current affairs updates, and important job-related information, all delivered by the country’s best teachers and industry experts.
If you also want to stay one step ahead in the race to success, subscribe to MADE EASY on YouTube and stay connected with us on social media.
MADE EASY — where preparation happens with confidence.
MADE EASY is a well-organized institute, complete in all aspects, and provides quality guidance for both written and personality tests. MADE EASY has produced top-ranked students in ESE, GATE, and various public sector exams. The publishing team regularly writes exam-related blogs based on conversations with the faculty, helping students prepare effectively for their exams.
