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What are the following terms- eignvalues of matrix-entries of matrix-equality of matrix-matrix of groups-identity of matrix-inverse matrix-multiplication of matrices?

In: Algebra, Abstract Algebra [Edit categories]

Answer:

First, a small note: an m-by-n or m x n matrix has m rows and n columns.

The eigenvalues λ of a matrix A are scalars such that Ax = λx for some nonzero x vector.
The entries a_ij of a matrix A are the numbers contained within the matrix, each with a unique position of the i^th row and j^th column.
'Equality' in matrices has the same definition as for the rest of mathematics.
A matrix of groups is a matrix whose entries are members of a group, often with specific entries in certain positions.
The matrix identity I_n is that square n by n matrix whose entries a_ij are 1 if i = j, and 0 if i ≠ j.
The inverse of a square matrix A is the square matrix B such that AB = I_n, denoted by B = A^-1.

Matrix multiplication is the act of combining two matrices, the p-by-q A = (a_ij) and the q-by-r B = (b_ij) to form the new matrix p-by-r C = (c_ij) such that c_ij = Σa_ikb_kj, where 1 ≤ k ≤ q. This is denoted by C = AB. Note that matrix mulplication is not commutative, i.e. AB does not necessarily equal BA; the order of the components is important and must be maintained to achieve the result. Note also that although p does not need to equal r, q must be the same in each matrix.

First answer by 3u8rbba98edy2. Last edit by 3u8rbba98edy2. Contributor trust: 742 Question popularity: 1 [recommend question].

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What is the -matrix of groups-identity of matrix-inverse matrix-multiplication of matrices?
The eigen values of a matirx are the values L such that Ax = Lx where A is a matrix, x is a vector, and L is a constant. The vector x is known as the eigenvector.
How are the inverse matrix and identity matrix related?
If an identity matrix is the answer to a problem under matrix multiplication, then each of the two matrices is an inverse matrix of the other.
Is Inverse of the inverse matrix the original matrix?
Let A by an nxn non-singular matrix, then A-1 is the inverse of A. Now (A-1 )-1 =A So the answer is yes.
How do you find original matrix from its inverse matrix?
To find the original matrix of an inverted matrix, simply invert it again. Consider A^-1^-1 = A^1 = A
Is the scalar matrix is always a identity matrix?
No. A scalar matrix is a diagonal matrix whose main diagonal elements are the same. Only if the diagonal elements are all 1 is it an identity matrix.