Matrix theory basics: several operations on the 2.3 square matrix

Section 3 arithmetic operations of phalanx

 

I. phalanx power

The k-th power of matrix A is counted as Ak.

Rules

Matrix Multiplication is suitable for combination laws, so the calculation of phalanx meets the following calculation rules:

Example 9

Solution:

Ii. Determining the phalanx

Definition 6NSquare MatrixAIs called a square matrix.AIs counted as |A| Or detA.

Example 10

Solution:

Note: The Order n square matrix is a number table, and the Order n determines a number.

 

The determining factor of the square matrix satisfies the following rules:

(1) |At| = |A|;

(2) |La| =LN|A|;

(3) |AB| = |A|B|, Similarly, |Ba| = |B|A|

From (3) We can see that for the n-order Determinant A and B, AB is generally equal to BA, but | BA | = | AB |.

Example 11

Solution:

It can also be interpreted

Define 7 square arrays of order nA|A| Algebra of each elementAIJThe transpose matrix of the n-order matrixA.

Note: | A | the algebraic remainder of each element in row I is located in column I (I = 1, 2 ,..., N ).

Solution:

We can see from the theorem in the Determinant

So

Likewise verifiable

The conclusion is as follows:

Solution:

 

 

 

Iii. Inverse of phalanx

Definition 8NLevel MatrixA, If yesNLevel MatrixB, MakingAB=Ba=En,

Then MatrixAIs a reversible or non-singular matrix, andBIsA.

AThe inverse array B of is counted as B =A-1.

Reversible matrices are mutual, B is the inverse matrix of A, and A is also the inverse matrix of B, and A and B are mutually inverse matrices.

That is, ifAB=Ba=E
, ThenB=A-1,A=B-1.

For example, for a second-order square matrix and a second-order square matrix

A and B are reversible matrices, and they are inverse matrices.

The inverse matrix has the following properties:

(1) uniqueness: If the MatrixAIs reversible, soAIs unique.

Proof: Because ifBN andCnAllAThe inverse matrix

AB=BA=E,
AC=CA=E,

ThereforeB=BE=B(AC) = (AB)C=EC=C

That isB=C

Therefore, the inverse matrix is unique.

(2) IfAReversible, thenA-1 is also reversible, and (A-1)-1 =A;

(3) IfAReversible, numberKLimit 0, thenKAReversible, and;

Proof:

(4) IfAReversible, thenATAnd (AT)-1 = (A-1)T
.

Proof: Because

AT(A-1)T= (A-1A)T=ET=E.

(A-1)T
= (A-1)T=ET=E.

So (AT)-1 = (A-1)T
.

(5) IfA,BFor the same-order reversible matrixABAnd (AB)-1 =B-1A-1.

This is because

(AB)(B-1A-1) =A(BB-1)A-1 =AEA-1 =AA-1 =E.

(B-1A-1 )(AB) =B-1 (A-1A)B
=B-1EB=B-1B=E.

So (AB)-1 =B-1A-1.

This property can also be extended to A finite matrix, such as A, B ,..., C is a reversible matrix.

(AB... C)-1 = C-1... B-1A-1.

Theorem 1 it is necessary and sufficient for n-order square matrix A to be reversible. The condition is | A | ≠ 0, and

CertificateNecessity:

A n-order matrix A-1 is known to be reversible, making

AA-1 = A-1A = En,

Because | AA-1 | = | A-1 | A | = | En | = 1,

So |A| Limit 0,

The property AA * = A * A = | A | E,

D,

Likewise

Therefore, according to the definition of the inverse arrayAReversible and

.

Adequacy (omitted)

In combination, the matrixAReversible license |A| Limit 0; ifAReversible, then

Theorem 1 not only provides the impulsive conditions for the existence of the inverse matrix, but also provides A method for finding the inverse matrix, that is, first finding the determinant of the known square matrix | A |, if |A| Limit 0. We can see that A is reversible and further calculated.

Solution:

The reversible nature of matrices can solve many complex problems.

For example, set a Linear EquationsAx = B,If A is A n-order reversible matrix, then the two sides of the equation are left by the A-1

A-1Ax =A-1B

ThenX =A-1BLinear EquationsAx = B.

Solution: known Coefficient Matrix

Therefore, a is a third-order reversible matrix.