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systems of linear equations & matrices

This document provides an overview of matrices and matrix operations. It defines what a matrix is and discusses matrix order and elements. It then covers basic matrix operations like scalar multiplication, addition, and multiplication. It introduces the concepts of transpose, special matrices like diagonal and triangular matrices, and the null and identity matrices. The document aims to define fundamental matrix concepts and arithmetic operations.

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SYSTEM OF LINEAR EQUATIONS & MATRICES
What is a matrix?  A matrix is a rectangular array of elements  The elements may be of any type (e.g. integer, real, complex, logical, or even other matrices).  In this course we will only consider matrices that have integer, real, or complex elements. −5 0 1 2 3 −4 −9 2 3 1 4 2      
Order of matrices…  Order 4 × 3:  Order 3 × 4: 3 columns→ 4 rows ↓ −5 0 1 2 3 −4 −9 2 6 3 1 4             4 columns→ 3 rows ↓ −5 0 1 2 3 −4 −9 2 3 1 4 2          
Specifying matrix elements aij denotes the element of the matrix A on the ith row and jth column. A= column j → row i ↓ −5 0 1 2 3 −4 −9 2 6 3 1 4         • a12 = 0 • a21 = 2 • a23 = -4 • a32 = 2 • a41 = 3 • a43 = 4
Matrix operations: scalar multiplication  Multiplying an m × n matrix by a scalar results in an m × n matrix with each of its elements multiplied by the scalar.  e.g.             −−− −− −− − =             − − − −             − − − =             − − − 826 12418 864 2010 413 629 432 105 2 1239 18627 1296 3015 413 629 432 105 3
Matrix operations: addition…  Adding or subtracting an m × n matrix by an m × n matrix results in an m × n matrix with each of its elements added or subtracted.  e.g.             − − − −−− =             − − −− −             − − −             − − − − =             − − −− +             − − − 431 8112 113 136 024 213 541 231 413 629 432 105 417 436 971 334 024 213 541 231 413 629 432 105
…Matrix operations: addition  Note that matrices being added or subtracted must be of the same order.  e.g. invalid! 113 201 413 629 432 105 =     − +             − − −
Matrix operations: multiplication…  Multiplying an m × n matrix by an n × p matrix results in an m × p matrix             ••••• ••••• ••••• ••••• ↓ → =           ••••• ••••• ••••• ↓ →             ••• ••• ••• ••• ↓ → wsrom columnsp wsron columnsp wsrom columnsn
…Matrix operations: multiplication…  Example 1… −1 0 2 3 1 1       2 −1 3 −2 1 −4           = 0      (−1×2)+(0×3)+(2×1)=0 −1 0 2 3 1 1       2 −1 3 −2 1 −4           = 0 −7      (−1×−1)+(0×−2)+(2×−4)=−7 −1 0 2 3 1 1       2 −1 3 −2 1 −4           = 0 −7 10       (3×2)+(1×3)+(1×1)=10 −1 0 2 3 1 1       2 −1 3 −2 1 −4           = 0 −7 10 −9       (3×−1)+(1×−2)+(1×−4)=−9
…Matrix operations: multiplication…  Example 2  i.e. the number of columns in the first matrix must equal the number of rows in the second matrix! invalid!=     −           − − − 113 201 124 862 351
…Matrix operations: multiplication…  Matrix multiplication is NOT commutative  In general, if A and B are two matrices then A B ≠ B A  i.e. the order of matrix multiplication is important!  e.g.       =                  =            10 22 10 02 10 21 10 42 10 21 10 02
…Matrix operations: transpose…  If B = AT , then bij = aji  i.e. the transpose of an m × n matrix is an n × m matrix with the rows and columns swapped.  e.g.           − −− =             − − − 4641 1230 3925 413 629 432 105 T [ ]3925 3 9 2 5 −−=             − − T
…Matrix operations: transpose…  (A B)T = BT AT  Note the reversal of order.  Justification (not a proof): e.g. if A is 3 × 2 and B is 2 × 4 then AT is 2 × 3 and BT is 4 × 2 so AT BT cannot be multiplied but BT AT can be multiplied.
Special matrices: row and column  A 1 × n matrix is called a row matrix. e.g.  An m × 1 matrix is called a column matrix. e.g. 1 columns→ 3 rows ↓ 3 −1 5           6 columns→ 1 rows ↓ 2 1 1 −2 1 −5[ ]
Special matrices: square  An n × n matrix is called a square matrix. i.e. a square matrix has the same number of rows and columns. e.g. [ ]             −− −−           −−       − 4705 7350 0523 5031 211 010 210 21 32 1
Special matrices: diagonal  A square matrix is diagonal if non-zero elements only occur on the leading diagonal. i.e. aij = 0 for i ≠ j e.g.  Premultiplying a matrix by a diagonal matrix scales each row by the diagonal element.  Postmultiplying a matrix by a diagonal matrix scales each column by the diagonal element. [ ]                             − 4000 0300 0020 0001 200 010 000 20 02 1
Special matrices: triangular  A lower triangular matrix is a square matrix having all elements above the leading diagonal zero. e.g.  An upper triangular matrix is a square matrix having all elements below the leading diagonal zero. e.g.             − − 4705 0350 0023 0001 1 2 0 −1    
Special matrices: null  The null matrix, 0, behaves like 0 in arithmetic addition and subtraction.  Null matrices can be of any order and have all of their elements zero. [ ] [ ]             =                 = = = 0000 0000 0000 0000 00 00 00 00 00 000 0 00 0 0
Special matrices: identity…  The identity matrix, I, behaves like 1 in arithmetic multiplication.  Identity matrices are diagonal. They have 1s on the diagonal and 0s elsewhere. e.g.  In the world of the matrix the identity truly is ‘the one’. [ ]             =           =       == 1000 0100 0010 0001 100 010 001 10 01 1 II II
…Special matrices: identity  The identity matrix multiplied by any compatible matrix results in the same matrix. i.e. I A = A e.g.  Any matrix multiplied by a compatible identity matrix results in the same matrix. i.e. A I = A e.g.  Multiplication by the identity matrix is thus commutative.       =            51 13 51 13 10 01       =            51 13 10 01 51 13
Determinant of a 2×2 matrix…  Matrices can represent geometric transformations, such as scaling, rotation, shear, and mirroring.  2 × 2 matrices can represent geometric transformations in a 2–dimensional space, such as a plane.  Determinants of 2 × 2 matrices give us information about how such transformations change the area of shapes.  Determinants are also useful to define the inverse of a matrix.
…Determinant of a 2×2 matrix…  The determinant of a 2 × 2 matrix is the product of the 2 leading diagonal terms minus the product of the cross- diagonal.  i.e. if A is a 2 × 2 matrix, then the determinant of A is denoted by det(A) = |A| = a11 a22 – a21 a12  e.g. det 3 −1 2 6      = 3 −1 2 6 = 3×6( )− 2×−1( )=20 det −2 5 1 3      = −2 5 1 3 = −2×3( )− 1×5( )=−11
…Determinant of a 2×2 matrix  |A B| = |A| |B|  Note the order is not important.  Justification (not a proof): We will shortly see that A and B can represent geometric transformations, and A B represents the combined transformation of B followed by A. The determinant represents the factor by which the area is changed, so the combined transformation changes area by a factor |A B|. Looking at the individual transformations, the area of the first is changed by a factor |B|, and the second by |A|. The overall transformation is thus changed by a factor |B| |A|, which is the same as |A| |B|.
Inverse of a matrix  In arithmetic multiplication the inverse of a number c is 1/c since c × 1/c = 1 and 1/c × c = 1  For matrices the inverse of a matrix A is denoted by A-1  A A-1 = I A-1 A = I where I is the identity matrix.  Multiplication of a matrix by its inverse is thus commutative.  We shall only consider the inverse of 2 × 2 matrices.
Inverse of a 2×2 matrix…  The inverse of a 2 × 2 matrix A is given by  Note: The leading term is 1/determinant; The diagonal elements are swapped; The cross-diagonal elements change their sign. a11 a12 a21 a22       −1 = 1 a11 a12 a21 a22 a22 −a12 −a21 a11      
…Inverse of a 2×2 matrix…  Example 1  Note that A A-1 = I (right inverse) and A-1 A = I (left inverse)       − =      − − =      − − 11 24 6 1 11 24 41 21 1 41 21 1       =      −       − 10 01 11 24 6 1 41 21       =      −      − 10 01 41 21 11 24 6 1
…Inverse of a 2×2 matrix…  Example 2  Note that the determinant is zero so the inverse does not exist for this matrix.  Matrices with zero determinant can have no inverse.  Such matrices are called singular. 2 2 3 3       −1 = 1 2 2 3 3 3 −2 −3 2      = 1 0 3 −2 2 2      = invalid!
…Inverse of a 2×2 matrix…  (A B)-1 = B-1 A-1  Note the reversal of order.  Justification (not a proof): B-1 A-1 A B = B-1 (A-1 A) B = B-1 B = I so B-1 A-1 is the inverse of A B i.e. (A B)-1 = B-1 A-1
29 Matrix Solution of Linear Systems When solving systems of linear equations, we can represent a linear system of equations by an augmented matrix, a matrix which stores the coefficients and constants of the linear system and then manipulate the augmented matrix to obtain the solution of the system. Example: x + 3y = 5 2x – y = 3 1 3 5 2 1 3     −  The augmented matrix associated with the above system is
30 Generalization  Linear system:  Associated augmented matrix: 11 12 1 21 22 2 a a k a a k       2222121 1212111 kxaxa kxaxa =+ =+
Operations that Produce Row-Equivalent Matrices  1. Two rows are interchanged:  2. A row is multiplied by a nonzero constant:  3. A constant multiple of one row is added to another row: i jR R↔ i ikR R→ j i ikR R R+ →
Augmented Matrix Method Example 1 Solve x + 3y = 5 2x – y = 3 1. Augmented system 2. Eliminate 2 in 2nd row by row operation 3. Divide row two by -7 to obtain a coefficient of 1. 4. Eliminate the 3 in first row, second position. 5. Read solution from matrix  : 1 2 2 2 2 1 1 1 3 5 2 1 3 2 1 3 5 0 7 7 / 7 1 3 5 0 1 1 3 10 2 2, 1;(2,1) 01 1 R R R R R R R x y     −  − + →     − −  − → →       − + → →   → = =    R2
Augmented Matrix Method Example 2 Solve x + 2y = 4 x + (1/2)y = 4 1. Eliminate fraction in second equation by multiplying by 2 2. Write system as augmented matrix. 3. Multiply row 1 by -2 and add to row 2 4. Divide row 2 by -3 5. Multiply row 2 by -2 and add to row 1. 6. Read solution : x = 4, y = 0 7. (4,0) 2 4 1 4 2 8 2 1 2 4 2 1 8 1 2 4 0 3 0 1 2 4 0 1 0 1 0 4 0 1 0 x y x y x y + = + = → + =   →      →  −    →      →   
Augmented Matrix Method Example 3 Solve 10x - 2y = 6 -5x + y = -3 1. Represent as augmented matrix. 2. Divide row 1 by 2 3. Add row 1 to row 2 and replace row 2 by sum 4. Since 0 = 0 is always true, we have a dependent system. The two equations are identical, and there are infinitely many solutions. 10 2 6 5 1 3 5 1 3 5 1 3 5 1 3 0 0 0  −    − −   −    − −   −     
Augmented Matrix Method Example 4  Solve  Rewrite second equation  Add first row to second row  The last row is the equivalent of 0x + 0y = -5  Since we have an impossible equation, there is no solution. The two lines are parallel and do not intersect. 5 2 7 5 1 2 x y y x − = − = + 5 2 7 5 2 2 5 2 7 5 2 2 5 2 7 0 0 5 x y x y − = −  → − + =   − −  →  −   − −    − 
Possible Final Matrix Forms for a Linear System in Two Variables Form 1: Unique Solution (Consistent and Independent) 1 0 0 1 m n       Form 2: Infinitely Many Solutions (Consistent and Dependent) 1 0 0 0 m n      Form 3: No Solution (Inconsistent) 1 0 0 m n p      
systems of linear equations & matrices

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In this document
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Introduction to systems of linear equations and defining matrix, including its order and element specification.

Introduction to basic matrix operations: scalar multiplication and addition; condition of matrix order in addition.

Matrix multiplication rules, properties, and examples; emphasizing order and non-commutativity.

Definition of matrix transpose, its properties, and implications in matrix dimensions.

Definitions and examples of special types of matrices: row, column, square, diagonal, triangular, null, and identity matrices.

Calculating determinants of 2x2 matrices, their geometric interpretations, and properties associated with transformations.

Concept and calculation of inverses for 2x2 matrices, including conditions for existence and order reversal in multiplication.

Using augmented matrices to solve linear systems, demonstration through various examples, including unique, infinite, and no solution cases.

systems of linear equations & matrices

  • 1.
  • 2.
    What is amatrix?  A matrix is a rectangular array of elements  The elements may be of any type (e.g. integer, real, complex, logical, or even other matrices).  In this course we will only consider matrices that have integer, real, or complex elements. −5 0 1 2 3 −4 −9 2 3 1 4 2      
  • 3.
    Order of matrices… Order 4 × 3:  Order 3 × 4: 3 columns→ 4 rows ↓ −5 0 1 2 3 −4 −9 2 6 3 1 4             4 columns→ 3 rows ↓ −5 0 1 2 3 −4 −9 2 3 1 4 2          
  • 4.
    Specifying matrix elements aijdenotes the element of the matrix A on the ith row and jth column. A= column j → row i ↓ −5 0 1 2 3 −4 −9 2 6 3 1 4         • a12 = 0 • a21 = 2 • a23 = -4 • a32 = 2 • a41 = 3 • a43 = 4
  • 5.
    Matrix operations: scalarmultiplication  Multiplying an m × n matrix by a scalar results in an m × n matrix with each of its elements multiplied by the scalar.  e.g.             −−− −− −− − =             − − − −             − − − =             − − − 826 12418 864 2010 413 629 432 105 2 1239 18627 1296 3015 413 629 432 105 3
  • 6.
    Matrix operations: addition… Adding or subtracting an m × n matrix by an m × n matrix results in an m × n matrix with each of its elements added or subtracted.  e.g.             − − − −−− =             − − −− −             − − −             − − − − =             − − −− +             − − − 431 8112 113 136 024 213 541 231 413 629 432 105 417 436 971 334 024 213 541 231 413 629 432 105
  • 7.
    …Matrix operations: addition Note that matrices being added or subtracted must be of the same order.  e.g. invalid! 113 201 413 629 432 105 =     − +             − − −
  • 8.
    Matrix operations: multiplication… Multiplying an m × n matrix by an n × p matrix results in an m × p matrix             ••••• ••••• ••••• ••••• ↓ → =           ••••• ••••• ••••• ↓ →             ••• ••• ••• ••• ↓ → wsrom columnsp wsron columnsp wsrom columnsn
  • 9.
    …Matrix operations: multiplication… Example 1… −1 0 2 3 1 1       2 −1 3 −2 1 −4           = 0      (−1×2)+(0×3)+(2×1)=0 −1 0 2 3 1 1       2 −1 3 −2 1 −4           = 0 −7      (−1×−1)+(0×−2)+(2×−4)=−7 −1 0 2 3 1 1       2 −1 3 −2 1 −4           = 0 −7 10       (3×2)+(1×3)+(1×1)=10 −1 0 2 3 1 1       2 −1 3 −2 1 −4           = 0 −7 10 −9       (3×−1)+(1×−2)+(1×−4)=−9
  • 10.
    …Matrix operations: multiplication… Example 2  i.e. the number of columns in the first matrix must equal the number of rows in the second matrix! invalid!=     −           − − − 113 201 124 862 351
  • 11.
    …Matrix operations: multiplication… Matrix multiplication is NOT commutative  In general, if A and B are two matrices then A B ≠ B A  i.e. the order of matrix multiplication is important!  e.g.       =                  =            10 22 10 02 10 21 10 42 10 21 10 02
  • 12.
    …Matrix operations: transpose… If B = AT , then bij = aji  i.e. the transpose of an m × n matrix is an n × m matrix with the rows and columns swapped.  e.g.           − −− =             − − − 4641 1230 3925 413 629 432 105 T [ ]3925 3 9 2 5 −−=             − − T
  • 13.
    …Matrix operations: transpose… (A B)T = BT AT  Note the reversal of order.  Justification (not a proof): e.g. if A is 3 × 2 and B is 2 × 4 then AT is 2 × 3 and BT is 4 × 2 so AT BT cannot be multiplied but BT AT can be multiplied.
  • 14.
    Special matrices: rowand column  A 1 × n matrix is called a row matrix. e.g.  An m × 1 matrix is called a column matrix. e.g. 1 columns→ 3 rows ↓ 3 −1 5           6 columns→ 1 rows ↓ 2 1 1 −2 1 −5[ ]
  • 15.
    Special matrices: square An n × n matrix is called a square matrix. i.e. a square matrix has the same number of rows and columns. e.g. [ ]             −− −−           −−       − 4705 7350 0523 5031 211 010 210 21 32 1
  • 16.
    Special matrices: diagonal A square matrix is diagonal if non-zero elements only occur on the leading diagonal. i.e. aij = 0 for i ≠ j e.g.  Premultiplying a matrix by a diagonal matrix scales each row by the diagonal element.  Postmultiplying a matrix by a diagonal matrix scales each column by the diagonal element. [ ]                             − 4000 0300 0020 0001 200 010 000 20 02 1
  • 17.
    Special matrices: triangular A lower triangular matrix is a square matrix having all elements above the leading diagonal zero. e.g.  An upper triangular matrix is a square matrix having all elements below the leading diagonal zero. e.g.             − − 4705 0350 0023 0001 1 2 0 −1    
  • 18.
    Special matrices: null The null matrix, 0, behaves like 0 in arithmetic addition and subtraction.  Null matrices can be of any order and have all of their elements zero. [ ] [ ]             =                 = = = 0000 0000 0000 0000 00 00 00 00 00 000 0 00 0 0
  • 19.
    Special matrices: identity… The identity matrix, I, behaves like 1 in arithmetic multiplication.  Identity matrices are diagonal. They have 1s on the diagonal and 0s elsewhere. e.g.  In the world of the matrix the identity truly is ‘the one’. [ ]             =           =       == 1000 0100 0010 0001 100 010 001 10 01 1 II II
  • 20.
    …Special matrices: identity The identity matrix multiplied by any compatible matrix results in the same matrix. i.e. I A = A e.g.  Any matrix multiplied by a compatible identity matrix results in the same matrix. i.e. A I = A e.g.  Multiplication by the identity matrix is thus commutative.       =            51 13 51 13 10 01       =            51 13 10 01 51 13
  • 21.
    Determinant of a2×2 matrix…  Matrices can represent geometric transformations, such as scaling, rotation, shear, and mirroring.  2 × 2 matrices can represent geometric transformations in a 2–dimensional space, such as a plane.  Determinants of 2 × 2 matrices give us information about how such transformations change the area of shapes.  Determinants are also useful to define the inverse of a matrix.
  • 22.
    …Determinant of a2×2 matrix…  The determinant of a 2 × 2 matrix is the product of the 2 leading diagonal terms minus the product of the cross- diagonal.  i.e. if A is a 2 × 2 matrix, then the determinant of A is denoted by det(A) = |A| = a11 a22 – a21 a12  e.g. det 3 −1 2 6      = 3 −1 2 6 = 3×6( )− 2×−1( )=20 det −2 5 1 3      = −2 5 1 3 = −2×3( )− 1×5( )=−11
  • 23.
    …Determinant of a2×2 matrix  |A B| = |A| |B|  Note the order is not important.  Justification (not a proof): We will shortly see that A and B can represent geometric transformations, and A B represents the combined transformation of B followed by A. The determinant represents the factor by which the area is changed, so the combined transformation changes area by a factor |A B|. Looking at the individual transformations, the area of the first is changed by a factor |B|, and the second by |A|. The overall transformation is thus changed by a factor |B| |A|, which is the same as |A| |B|.
  • 24.
    Inverse of amatrix  In arithmetic multiplication the inverse of a number c is 1/c since c × 1/c = 1 and 1/c × c = 1  For matrices the inverse of a matrix A is denoted by A-1  A A-1 = I A-1 A = I where I is the identity matrix.  Multiplication of a matrix by its inverse is thus commutative.  We shall only consider the inverse of 2 × 2 matrices.
  • 25.
    Inverse of a2×2 matrix…  The inverse of a 2 × 2 matrix A is given by  Note: The leading term is 1/determinant; The diagonal elements are swapped; The cross-diagonal elements change their sign. a11 a12 a21 a22       −1 = 1 a11 a12 a21 a22 a22 −a12 −a21 a11      
  • 26.
    …Inverse of a2×2 matrix…  Example 1  Note that A A-1 = I (right inverse) and A-1 A = I (left inverse)       − =      − − =      − − 11 24 6 1 11 24 41 21 1 41 21 1       =      −       − 10 01 11 24 6 1 41 21       =      −      − 10 01 41 21 11 24 6 1
  • 27.
    …Inverse of a2×2 matrix…  Example 2  Note that the determinant is zero so the inverse does not exist for this matrix.  Matrices with zero determinant can have no inverse.  Such matrices are called singular. 2 2 3 3       −1 = 1 2 2 3 3 3 −2 −3 2      = 1 0 3 −2 2 2      = invalid!
  • 28.
    …Inverse of a2×2 matrix…  (A B)-1 = B-1 A-1  Note the reversal of order.  Justification (not a proof): B-1 A-1 A B = B-1 (A-1 A) B = B-1 B = I so B-1 A-1 is the inverse of A B i.e. (A B)-1 = B-1 A-1
  • 29.
    29 Matrix Solution ofLinear Systems When solving systems of linear equations, we can represent a linear system of equations by an augmented matrix, a matrix which stores the coefficients and constants of the linear system and then manipulate the augmented matrix to obtain the solution of the system. Example: x + 3y = 5 2x – y = 3 1 3 5 2 1 3     −  The augmented matrix associated with the above system is
  • 30.
    30 Generalization  Linear system: Associated augmented matrix: 11 12 1 21 22 2 a a k a a k       2222121 1212111 kxaxa kxaxa =+ =+
  • 31.
    Operations that Produce Row-EquivalentMatrices  1. Two rows are interchanged:  2. A row is multiplied by a nonzero constant:  3. A constant multiple of one row is added to another row: i jR R↔ i ikR R→ j i ikR R R+ →
  • 32.
    Augmented Matrix Method Example1 Solve x + 3y = 5 2x – y = 3 1. Augmented system 2. Eliminate 2 in 2nd row by row operation 3. Divide row two by -7 to obtain a coefficient of 1. 4. Eliminate the 3 in first row, second position. 5. Read solution from matrix  : 1 2 2 2 2 1 1 1 3 5 2 1 3 2 1 3 5 0 7 7 / 7 1 3 5 0 1 1 3 10 2 2, 1;(2,1) 01 1 R R R R R R R x y     −  − + →     − −  − → →       − + → →   → = =    R2
  • 33.
    Augmented Matrix Method Example2 Solve x + 2y = 4 x + (1/2)y = 4 1. Eliminate fraction in second equation by multiplying by 2 2. Write system as augmented matrix. 3. Multiply row 1 by -2 and add to row 2 4. Divide row 2 by -3 5. Multiply row 2 by -2 and add to row 1. 6. Read solution : x = 4, y = 0 7. (4,0) 2 4 1 4 2 8 2 1 2 4 2 1 8 1 2 4 0 3 0 1 2 4 0 1 0 1 0 4 0 1 0 x y x y x y + = + = → + =   →      →  −    →      →   
  • 34.
    Augmented Matrix Method Example3 Solve 10x - 2y = 6 -5x + y = -3 1. Represent as augmented matrix. 2. Divide row 1 by 2 3. Add row 1 to row 2 and replace row 2 by sum 4. Since 0 = 0 is always true, we have a dependent system. The two equations are identical, and there are infinitely many solutions. 10 2 6 5 1 3 5 1 3 5 1 3 5 1 3 0 0 0  −    − −   −    − −   −     
  • 35.
    Augmented Matrix Method Example4  Solve  Rewrite second equation  Add first row to second row  The last row is the equivalent of 0x + 0y = -5  Since we have an impossible equation, there is no solution. The two lines are parallel and do not intersect. 5 2 7 5 1 2 x y y x − = − = + 5 2 7 5 2 2 5 2 7 5 2 2 5 2 7 0 0 5 x y x y − = −  → − + =   − −  →  −   − −    − 
  • 36.
    Possible Final MatrixForms for a Linear System in Two Variables Form 1: Unique Solution (Consistent and Independent) 1 0 0 1 m n       Form 2: Infinitely Many Solutions (Consistent and Dependent) 1 0 0 0 m n      Form 3: No Solution (Inconsistent) 1 0 0 m n p      