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

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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:
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Calculating the rank of a matrix for determining
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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 3 = c 1 + c 2
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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.
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4. Will rank 3 if there is a square submatrix of order 3, such that its determinant is not zero.
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As all the determinants of the submatrices are zero has rank 3, then r (B) = 2.
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
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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.
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1. We find the rank of the matrix of coefecientes.
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2. We find the rank of the augmented matrix.
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3. We apply Rouche's theorem
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4. We solve the system compatible determined by Cramer's rule (also can be solved by the Gauss).
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Calculating the inverse matrix
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Calculating the inverse matrix
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1. We calculate the determinant of the matrix, where the determinant is null or the array will not reverse.
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2. We find the attached matrix, which is one in which every element is replaced his deputy po

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> 3. We calculate the transpose of the matrix attached.
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4. The inverse matrix is equal to the inverse of the value of its determinant for the matrix transpose of the enclosed.
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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
IX = A -1 C
X = A -1 C
3rd case
XA = C
If there is the inverse of A, | A |? 0
AA -1 X = CA -1
C IX = A -1
X = CA -1
4th case
AX + BX = C
(A + B) X = C
(A + B) -1 (A + B) X = (A + B) -1 C
IX = (A + B) -1 C
X = (A + B) -1 C
Multiplication Arrays
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Multiple rows of columns

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