Similarity Of 2 Matrices

Two matrices are said to survive the same, if they accept the same gild together with the elements of the ii matrices are the same.

Example:
Find the x, y, z, a, b, together with x values of the equality of the ii matrix below!

Answer:

1. Element row i column i (a₁₁)
2x = 4
x = 2

2. Element row i column three (a₁₃)
2 + x = y
y = 2 + 2
y = 4

3. Element row 2 columnt i (a₂₁)
z = 3y
z = 3(4)
z = 12

4. Elemen row 2 columnt 2 (a₂₂)
a + i = 4z
a + i = 4(12)
a = 48 - 1
a  = 47

5. Element row three columnt i (a₃₁)
b = a + 5
b = 47 + 5
b = 52

6. Elemen row three columnt 2 (a₃₂)
(1/2)c = b - 2
(1/2)c = 52 - 2
(1/2)c = 50
(1/2)c x 2 = fifty x 2
c = 100

So from the matrix is obtained:
a = 47
b = 52
c = 100
x = 2
y = 4
z = 12

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Transpose Matrix

Transpose matrix Influenza A virus subtype H5N1 = (a_ij) amongst the gild 1000 x n is written A^T = (a_ji) in addition to has the gild 1000 x n. The elements of the matrix row AT are obtained from elements of the matrix column Influenza A virus subtype H5N1 in addition to vice versa.

Example:

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Summation Together With Reduction Matrix

Two matrices A together with B tin locomote summed or used a subtraction functioning if the companionship (row x column) of the matrix code is the same. The lawsuit of the amount or divergence is obtained past times summing or subtracting the elements that prevarication inside the 2 matrices.

Example:
Is known:

Answer:

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Determinant Of Matrix

1. Determinant of the Two Order Matrices

Ex:

Then the determinant of A(det(A)) is:

Example:
Determine the determinant of the matrix below!

Answer:

2. Determinant of the Three Order Matrices

Suppose a foursquare matrix alongside an gild non given is given below:

Then the determinant of the matrix is:

There are many ways that tin flame hold upward used to calculate the determinant of matrices alongside 3 x 3 orders, exactly the close widely used is past times using sarrus rules. With the steps every bit follows:

• Put the commencement in addition to minute columns to the correct of the vertical trouble of the determinant.
• The release of times the elements located on the primary diagonal alongside the production times the elements that are parallel to the primary diagonal inwards the correct direction, therefore subtracted past times the production times the elements lying parallel to the side diagonal.

Consider the system to calculate using the next sarrus rules:

Example:
Determine the determinant of the matrix below!

Answer:

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Matrix Multiplication

1. Multiplication Matrix alongside Scalar

For Example k is a scalar as well as A is a matrix, as well as therefore kA is a matrix obtained yesteryear multiplying every chemical ingredient of the matrix Influenza A virus subtype H5N1 yesteryear a scalar.

Example:
Is known:

b. Matrix Multiplication alongside Matrix

Two matrices A alongside ordoo m 10 n as well as matrix B alongside the social club n 10 p, the production times betwixt A as well as B is a matrix C = A. B, which has an social club of m 10 p, is obtained yesteryear multiplying each row matrix chemical ingredient A alongside matrix chemical ingredient B.

If the matrix A has an social club of m 10 n as well as B has an social club p 10 q where n is non equal to p as well as therefore A. B is undefined. Consider the illustrations of domino cards inwards the figure below for the multiplication of 2 matrices of each social club of 2 10 4 as well as 4 10 1.

Example:
Is Known:

Determine A. B!

Answer:
Matrix A has 2 10 2 as well as B orders 2 10 3, the production of A. B is an orderly matrix 2 10 3. Consider the instance below!

is the 1st row entry as well as the 2d column of the matrix Influenza A virus subtype H5N1 obtained yesteryear multiplying the 1st row chemical ingredient of the left matrix (matrix A) alongside the 2d column elements of the matrix to the correct (matrix B) as well as therefore summing it upwards . And therefore on to fill upwards the boxes.

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Matrix Inward Math

Definition of Matrix

Matrix is an organization of elements or entries inward rectangular cast arranged inward rows together with columns.

Matrix Order

The social club or size of the matrix is the publish of row elements followed past times the publish of columns. A_mxn agency matrix H5N1 has an social club of m x n, important that the metric has one thousand rows together with n pieces of column.

Example:

Answer:
Matrix A consists of 2 rows together with 4 columns, the matrix H5N1 is 2 x 4, or A_mxn.

Types of Matrices

3.1 Zero Matrix

The nothing matrix is a matrix whose all elements are zero.

Example:

3.2 Column Matrix

The column matrix is a matrix consisting of but 1 column.

Example:

3.3 Row Matrices

Row matrix is a matrix consisting of but 1 line.

Example:

3.4 Square Matrix

The foursquare matrix is a matrix whose publish of rows is equal to the publish of columns.

Example:

3.5 Diagonal Matrix

The diagonal matrix is a matrix whose entire chemical cistron is nothing except that the primary diagonal is non e'er zero.

Example:

3.6. Triangle Matrix

Triangle matrix consists of ii kinds, such as:

3.6.1 Top Triangle Matrix
The upper triangular matrix is a matrix whose elements below the primary diagonal are but zero.

Example:

3.6.2 Bottom Triangle Matrix
The lower triangular matrix is a matrix whose elements to a higher house the primary diagonal are all zero.

Example:

3.7 Identity Matrix

The identity matrix is a foursquare matrix that all elements on the diagonal are primarily 1 together with the other is zero.

Example:

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Minor, Cofactor, Too Adjoin Matrix

If Influenza A virus subtype H5N1 is a rectangular matrix, the youngster entries or elements aij are expressed past times M_ij in addition to are defined equally the determinant of the submatrix that resides afterwards the ith row in addition to the jth column are crossed from A. The numbers (-1)^{i + j} Mij are expressed past times C_ij called aij cofactor entries.

If Influenza A virus subtype H5N1 is whatever rectangular matrix (n 10 n) in addition to C_ij is a cofactor a_ij, in addition to then the matrix:

is called the cofactor matrix of A. Transpose This matrix is called the adjoint of A in addition to is denoted past times Adj(A).

Example:
Determine the minor, cofactor, cofactor matrix, in addition to adjoin of:

Answer:
Minor from matrix A is:
M₁₁ = 4
M₁₂ = 5
M₂₁ = 1
M₂₂ = -2

The cofactor of the matrix Influenza A virus subtype H5N1 is:
C₁₁ = (-1)¹⁺¹ M₁₁ = (1)4 = 4
C₁₂ = (-1)¹⁺² M₁₂ = (-1)5 = -5
C₂₁ = (-1)²⁺¹ M₂₁ = (-1)(1) = -1
C₂₂ = (-1)²⁺² M₂₂ = (1)(-2) = -2

The cofactor matrix is:

The adjoin of the cofactor matrix is the transpose of the cofactor matrix, therefore that:

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