Diagonal element and main diagonal

Several special square arrays

n-dimensional row vector

m-dimensional column vector

Prerequisites for matrix equality

addition

Multiply numbers

Premise: Number of rows in A = number of columns in B

The sum of the products of the corresponding elements of the i-th row of A and the j-th column of B

operational laws

Notice

Prerequisite: Square Array

Similar to the operation of exponentiation

A raised to the 0th power = E

Polynomial of degree m of A

Replace all rows with corresponding columns

operational laws

Transposed matrix = original matrix

Transposed matrix =-original matrix

Prerequisite: If it is a square matrix

nature

Inverse of A=A adjoint/determinant of A

The adjoint inverse of A = A/|A|

Determinant of A adjoint =|A|^(n-1)

The inverse of the inverse is the original matrix

The inverse of kA is the inverse of k/k A

The inverse determinant of A = one-tenth of the determinant of A

The inverse of the transpose of A = the transpose of the inverse of A

The inverse of AB multiplied by = the inverse of B times the inverse of A

same side elimination law

Solving a system of linear equations using the inverse of a matrix: x = inverse of A multiplied by b

A determinant, a number

There is a non-zero subformula of order r in A, and all subformulas of order (r-1) are zero, then the rank of A is r

It is stipulated that r(0)=0

r(A) is less than the number of rows or columns of A, whichever is smaller