Mathematics Notes Chapter 3_Matrices
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Transcript of Mathematics Notes Chapter 3_Matrices
CHAPTER 2: MATRICES
CHAPTER 2: MATRICES
CHAPTER 2: MATRICES
DQT 101
2.1Introduction
If a matrix has m rows and n columns, then the matrix is called matrix. Normally, matrix can be written as where is denotes the elements i-th row and j-th column. If for , then the elements is called the leading diagonal of matrix A. More generally, matrix A can be written as
Example 2.1Let .Determinei) order of matrix Aii) elements of the leading diagonal of matrix Aiii) elements , and of matrix ASolutioni) order of matrix A is .ii) 8, 9 and 3 are elements of the leading diagonal.
iii) , and .2.2Type of Matrices
Example 2.2Determine the following matrices are diagonal matrices or not.
, ,
SolutionMatrix A is not a diagonal matrix because .
Matrix B is a diagonal matrix because and for .
Matrix C is not a diagonal matrix because .
Example 2.3Determine the following matrices are scalar matrix or not.
, ,
SolutionMatrix A is a scalar matrix where .
Matrix B is a scalar matrix where .
Matrix C is not a scalar matrix because the diagonal elements are not equal.
Example 2.4
Determine the following matrices are identity matrix or not.
, ,
SolutionMatrix A is a identity matrix where .
Matrix B is not a identity matrix because the matrix is a matrix.Matrix C is not a identity matrix because the matrix is a matrix.
Example 2.5
Determine the following matrices are zero matrices or not.
, ,
SolutionMatrices A, B and C are zero matrices because all the elements of the matrices are zero.
Example 2.6
Determine the negative matrices of A and Bi)
ii)
Solutioni) , hence
ii) , hence
Example 2.7
Determine the following matrices are upper triangular matrix or lower triangular matrix.i)
ii)
Solutioni) is a upper triangular matrix.ii) is a lower triangular matrix.
Example 2.8
Determine the transpose of the following matrices.i)
ii)
Solutioni)
ii)
Example 2.9Let, find. Show that the matrix of A is a symmetric matrix?
Solution
. Hence, A is a symmetric matrix.
Example 2.10Let , find . Shows that the matrix of A is a skew symmetric matrix?
Solution
. Hence, A is a skew symmetric matrix.
Example 2.11
For the following matrices, determine the matrices to be in row echelon form or not. If the matrices arent in row echelon form, give a reason.
i)
ii) iii)
Solutioni) Matrix A isnt in row echelon form because the number 1 in second
rows appears in the same column.ii) Matrix B is a row echelon form.
iii) Matrix C is a row echelon form.
Example 2.12
For the following matrices, determine the matrices to be in reduced row echelon form or not. If the matrices arent in reduced row echelon form, give a reason.
i)
ii)
iii)
Solutioni) Matrix A is in reduced row echelon form.
ii) Matrix B is in reduced row echelon form.
iii) Matrix C isnt in reduced row echelon form because .
2.3Matrix OperationsExample 2.13Given and . If P = Q, find the value of x, y and z.
SolutionGiven and the solution is x = 4, y = 5 and z = 0.
Example 2.14
Given , and.
Findi) A + B
ii) A + Ciii) B + C
Solutioni)
ii) A + C (is not possible)iii) B + C (is not possible)Note: It should be noted that the addition of the matrices A and B is defined only when A and B have the same number of rows and the same number of columns
Example 2.15
Given , and .
Findi) A B
ii) B A
iii) B C
Solutioni)
ii)
iii) B C (is not possible)Note: It should be noted that the subtraction of the matrices A and B is defined only when A and B have the same number of rows and the same number of columns and.
Properties of Matrices Addition and SubtractionIf A, B and C are matrices, theni) A + B = B + A
ii) A + (B + C) = (A + B) + Ciii) A + 0 = A
iv) A + (A) = 0
v)(A + B)T = AT + BT
Example 2.16Given .
Findi) 3A
ii) -2ASolutioni)
ii)
Properties of Scalar MultiplicationIf A and B are both matrices, k and p are scalar, theni)
ii)
iii)
iv)(kA)T = kAT
Example 2.17
Given and .
Find
i) AB
ii) BA
Solutioni) AB =
EMBED Equation.3 =
ii) BA (is not possible)Properties of Matrix Multiplication
If A, B and C are matrices, I identity matrix and 0 zero matrix, theni) A(B + C) = AB + ACii)(B + C)A = BA + CAiii) A(BC) = (AB)C
iv)IA = AI = Av) 0A = A0 = 0
vi)(AB)T = BTAT2.4Determinants
Example 2.18
Find the determinant of matrix.Solution|A| =
Method of Calculation
- - - + + +
Example 2.19
Given , evaluate |P|.Solution
- - - + + +So that, |P| = 2(5)(3) + 1(2)(4) + 1(3)(3) 1(5)(4) 2(2)(3) 1(3)(3)
= 30 + 8 + 9 20 12 9
= 6
Note: Methods used to evaluate the determinant above is limited to only and matrices. Matrices with higher order can be solved by using minor and cofactor methods.2.5Minor, Cofactor and Adjoint
Example 2.20Let. Evaluate the following minorsi)
ii)
iii)
Solution
i)
ii)
iii)
Example 2.21
Let. Evaluate the following cofactorsi)
ii)
iii)
Solution
i)
ii)
iii)
Example 2.22
Find the determinant of by expanding along the second row.Solution
Expand by the second row,|A| =
= 0 + 1(1)4 + 0 + 3(1)6
= 1(1)44(1)5 + 3(1)61(1)4
= 4(3) + 3(3)
= 3
Properties of Determinanti)If A is a matrix, then |A| = |AT|
ii)If two rows (columns) of A are equal, the |A| = 0.
iii)If a row (column) of A consist entirely of zeros elements, then |A| = 0.
iv)If B is obtained from multiplying a row (column) of A by a scalar k, then |B| = k |A|.v)To any row(column) of A we can add or subtract any multiple of any other row (column) without changing |A|.vi)If B is obtained from A by interchanging two rows (columns), then |B| = |A|.
Example 2.23
Let , compute adj [A].Solution
= (1)1+1M11 = = 2, = (1)1+2 M12 = 1 = 20
= (1)1+3M13 = = 28, = (1)2+1 M 21 = 1 = 16
= (1)2+2M 22 = = 10, = (1)2+3M 23 = 1 = 14
= (1)3+1M 31 = = 18, = (1)3+2 M 32 = 1 = 0
= (1)3+3M 33 = = 12
We have Cij =
Then, adj [A] = [Cij]T =
2.6Inverses of MatricesIf AB = I, then A is called inverse of matrix B or B is called inverse of matrix A and denoted by A = B-1 or B = A-1.
Example 2.24Determine matrix is the inverse of matrix .Solution
Note that
EMBED Equation.3
EMBED Equation.3 Hence AB = BA = I, so B is the inverse of matrix A (B = A-1)
Example 2.25
Let, find A-1.SolutionA-1 = .
Example 2.26
Let, find A-1.SolutionFrom the example 2.23, adj [A] = and |A| = 60.
Then,
Characteristic of Elementary Row Operations (ERO)i)Interchange the i-th row and j-th row of a matrix, written as .
ii)Multiply the i-th row of a matrix by a nonzero scalar k, written as kbi.
iii)Add or subtract a constant multiple of i-th row to the j-th row, written as or
Example 2.27
By performing the elementary row operations (ERO), find the inverse of matrix .
Solution
Then, .2.7Solving the Systems of Linear Equation2.7.1Inversion MethodConsider the following system of linear equations with n equations and n unknowns.
.
.
.
The systems of linear equations can be written as a single matrix equation AX = B, that is
EMBED Equation.3 =
Here, A is a coefficients matrix, X is the vector of unknowns and B is a vector containing the right hand sides of the equations. The solution is obtained by multiplying both side of the matrix equation on the left by the inverse of matrix A:A-1AX = A-1B
IX = A-1B
X = A-1BExample 2.28
Solve the system of linear equations by using the inverse matrix method.
SolutionA = ,X = and B =
AX = B =
EMBED Equation.3 =
A-1 =
Then,
X = A-1B =
EMBED Equation.3 =
Hence, , and .2.7.2 Gaussian Elimination Consider the following system of linear equations with n equations and n unknowns.
.
.
.
The system of linear equations can be written in the augmented form that is [ A | B ] matrix and state the matrix in the following form :
By using elementary row operations (ERO) on this matrix such that the matrix A may reduce in the row echelon form (REF). It is called a Gaussian elimination process.Example 2.29
Solve the system of linear equations by using Gaussian elimination.
Solution
From the Gaussian elimination, we have
Then, the solution is, and .
2.7.3Gauss-Jordan EliminationConsider the following system of linear equations with n equations and n unknowns.
.
.
.
The system of linear equations can be written in the augmented form that is [ A | B ] matrix and state the matrix in the following form :
By using the elementary row operations (ERO) on this matrix such that the matrix A may reduce in the reduced row echelon form (RREF). The procedure to reduce a matrix to reduced row echelon form is called Gauss-Jordan elimination.
Example 2.30
Solve the system of linear equations by using Gauss-Jordan elimination.
Solution
From the Gauss-Jordan elimination, the solution of linear equation is , and .
2.7.4Cramers RuleIf AX = B is a system of n linear equations with n unknown such that , then the system has a unique solution; where is the matrix obtained by replacing the entries in the i-th column of A by the entries in the matrix B.
Example 2.31
Solve the system of linear equations by using Cramers Rule.2x + 4y + 6z = 18
4x + 5y + 6z = 24
3x + y 2z = 4
SolutionWrite a single matrix equation AX = B, that is
EMBED Equation.3 =
Then,
x = = = 4
y = = = 2
z = = = 3
CHAPTER 2 : MATRICES
Definition 2.1
A matrix is an array of real numbers in m rows and n columns
Definition 2.2
If A is an EMBED Equation.3 matrix, then A is called a square matrix when the number of row (i) is equal to the number of column (j).
Definition 2.3
An EMBED Equation.3 matrix is called a diagonal matrix if EMBED Equation.3 and EMBED Equation.3 for EMBED Equation.3 .
Definition 2.4
An EMBED Equation.3 matrix is called a scalar matrix if it is a diagonal matrix in which the diagonal elements are equal, that is EMBED Equation.3 and EMBED Equation.3 for EMBED Equation.3 where k is a scalar.
Definition 2.5
An EMBED Equation.3 matrix is called identity matrix if the diagonal elements are equal to 1 and the rest of elements are zero, that is EMBED Equation.3 and EMBED Equation.3 for EMBED Equation.3 . Identity matrix is denoted by EMBED Equation.3 .
Definition 2.6
An EMBED Equation.3 matrix is called a zero matrix if all the elements of the matrix are zero, that is EMBED Equation.3 and is written as EMBED Equation.3 .
Definition 2.7
A negative matrix of EMBED Equation.3 is denoted by -A where EMBED Equation.3 .
Definition 2.8
An EMBED Equation.3 matrix EMBED Equation.3 is called upper triangular matrix if EMBED Equation.3 for EMBED Equation.3 . It is called lower triangular matrix if EMBED Equation.3 for EMBED Equation.3 .
Definition 2.9
If EMBED Equation.3 is an EMBED Equation.3 matrix, then the tranpose of A, EMBED Equation.3 is the EMBED Equation.3 matrix defined by EMBED Equation.3 .
Definition 2.10
An EMBED Equation.3 matrix is called a symmetric matrix if EMBED Equation.3 .
Definition 2.11
An EMBED Equation.3 matrix with real entries is called a skew symmetric matrix if EMBED Equation.3 .
Definition 2.12
An EMBED Equation.3 matrix A is said to be in row echelon form (REF) if it satisfies the following properties :
All zero rows, if there any, appear at the bottom of the matrix.
The first nonzero entry from the left of a nonzero is a number 1. This entry is called a leading 1 of its row.
For each non zero row, the number 1 appears to the right of the leading 1 of the previous row.
Definition 2.13
An EMBED Equation.3 matrix A is said to be in reduced row echelon form (RREF) if it satisfies the following properties :
All zero rows, if there any, appear at the bottom of the matrix.
The first nonzero entry from the left of a nonzero is a number 1. This entry is called a leading 1 of its row.
For each non zero row, the number 1 appears to the right of the leading 1 of the previous row.
If a column contains a leading 1, then all other entries in the column are zero
Definition 2.14
Two matrices EMBED Equation.3 and EMBED Equation.3 are equal if EMBED Equation.3 and denoted by A = B.
Definition 2.15
If EMBED Equation.3 and EMBED Equation.3 are both EMBED Equation.3 matrices, then A + B is an EMBED Equation.3 matrix EMBED Equation.3 defined by EMBED Equation.3 .
Definition 2.16
If EMBED Equation.3 and EMBED Equation.3 are both EMBED Equation.3 matrices, then the A - B is an EMBED Equation.3 matrix EMBED Equation.3 defined by EMBED Equation.3 .
Definition 2.17
If EMBED Equation.3 is an EMBED Equation.3 matrix and k is a scalar, then the scalar multiplication is denoted kA where EMBED Equation.3
Definition 2.18
If A is an EMBED Equation.3 matrix and B is an EMBED Equation.3 matrix, then the product EMBED Equation.3 is the EMBED Equation.3 matrix.
Definition 2.19
If EMBED Equation.3 is a EMBED Equation.3 matrix, then the determinant of A denoted by det[A] = |A| and is given by |A| = EMBED Equation.3 .
Definition 2.20
If EMBED Equation.3 is a EMBED Equation.3 matrix, then the determinant of A is given by
|A|= EMBED Equation.3 .
Definition 2.21
Let EMBED Equation.3 be an EMBED Equation.3 matrix. Let EMBED Equation.3 be the EMBED Equation.3 submatrix of A is obtained by deleting the i-th row and j-th column of A. The determinant EMBED Equation.3 is called the minor of A.
Definition 2.22
Let EMBED Equation.3 be an EMBED Equation.3 matrix. The cofactor EMBED Equation.3 of EMBED Equation.3 is defined as EMBED Equation.3 .
Definition 2.23
If EMBED Equation.3 is cofactor of matrix A, then the determinant of matrix A can be obtained by
i) Expanding along the i-th row
EMBED Equation.3
OR
ii) Expanding along the j-th column
EMBED Equation.3
Definition 2.24
Let A is an EMBED Equation.3 matrix, then the adjoint of A is defined as
adj [A] = [Cij]T.
Definition 2.25
An EMBED Equation.3 matrix A is said to be invertible if there exist an EMBED Equation.3 matrix B such that AB = BA = I where I is identity matrix.
Theorem 1
If EMBED Equation.3 and EMBED Equation.3 , then A-1 = EMBED Equation.3 .
Theorem 2
If A is EMBED Equation.3 matrix and EMBED Equation.3 , then A-1 = EMBED Equation.3 adj [A].
Theorem 3
If augmented matrix [ A | I ] may be reduced to [ I | B] by using elementary row operation (ERO), then B is called inverse of A.
Theorem 4
The unique solution of the system of linear equation
EMBED Equation.3
EMBED Equation.3
with nonzero coefficient determinant is given by
EMBED Equation.3 and EMBED Equation.3
__________________________________________________________________
Universiti Malaysia Perlis 200844 Universiti Malaysia Perlis
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