# Matrices: multiplication, rank, determinant, inverse and Rouche-Frobenius theorem

Classified in Mathematics

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**System Types**

The systems of equations can be classified by the number of solutions that can arise. According to that case may have the following cases:

· **Incompatible system** if it has no solution.

· **Compatible system** if you have any solution in this case can also distinguish between:

or **compatible system determined** when it has a finite number of solutions. **indeterminate** or **compatible system** when it admits an infinite set of solutions.

Fitting and classification: **Calculating the rank of a matrix for determining ** 1. We can rule a line if:.

·

**All the coefficients are zeros.**

·

**There are two equal lines.**

•

**A line is proportional to another.**

•

**A line is a linear combination of others.**

Delete the third column because it is a linear combination of the first two:

**c**

2. We check if you have rank 1, for it must be satisfied that at least one array element is not zero and therefore its determinant is not zero.

| 2 | = 2 • 0

_{3}= c_{1}+ c_{2}2. We check if you have rank 1, for it must be satisfied that at least one array element is not zero and therefore its determinant is not zero.

| 2 | = 2 • 0

**3. Will rank 2 if there is any square submatrix of order 2, such that its determinant is not zero.**

4. Will rank 3 if there is a square submatrix of order 3, such that its determinant is not zero.

As all the determinants of the submatrices are zero has rank 3, then r (B) = 2.

4. Will rank 3 if there is a square submatrix of order 3, such that its determinant is not zero.

**5. If you have rank 3 and there is a submatrix of order 4, whose determinant is not zero, you have rank 4.**In this same way you work to check if you have range greater than 4.

Determinant

3x3

**Discussion of systems: Rouche-Frobenius theorem**

The necessary and sufficient condition for a system of m equations and n unknowns has a solution is that the **range of the coefficient matrix and the extended matrix are equal. ** ·

**R = r 'System Compatible.**

or

**r = r '= n Determined System Compatibility.**

or

**r = r '? No Compatible System Undetermined.**

·

**R? r 'incompatible systems.**

Study and resolve, if possible, the system:

using determinants and Rouche-Frobenius theorem.

1.

**We find the rank of the matrix of coefecientes.**

2.

**We find the rank of the augmented matrix.**

3. We apply Rouche's theorem

4.

**We solve the system compatible determined by Cramer's rule**(also can be solved by the Gauss).

**Calculating the inverse matrix **

Calculating the inverse matrix

1. We calculate the determinant of the matrix, where the determinant is null or the array will not reverse.

2. **We find the attached matrix, which is one in which every element is replaced his deputy po**

> 3. We calculate the transpose of the matrix attached.

4. The inverse matrix is equal to the inverse of the value of its determinant for the matrix transpose of the enclosed.

Matrix equations formulas

^{1st} case

X + B = C

X + B? B = C? B

X = C? B

2nd case

AX = C

If there is the inverse of A, **| A |? 0** **A ^{-1} AX = A ^{-1} C ** If there is the inverse of A,

IX = A ^{-1} C

X = A ^{-1} C

^{3rd} case

XA = C

**| A |? 0**

**AA**

AX + BX = C

(A + B) X = C

(A + B)

IX = (A + B)

X = (A + B)

Multiplication Arrays

^{-1}X = CA^{-1 }C IX = A^{-1 }X = CA^{-1 }4th caseAX + BX = C

(A + B) X = C

(A + B)

^{-1}(A + B) X = (A + B)^{-1}CIX = (A + B)

^{-1}CX = (A + B)

^{-1}CMultiple rows of columns