Matrix and Determinant - Transpose of Matrix and its Properties

  • The transpose of a matrix refers to interchange the rows and columns of a matrix.
  • It is denoted by A^T or A'
  • The transpose operation can be performed on any rectangular matrix.

Transpose of a Matrix (Example)

Consider the matrix A: [ A = \begin{pmatrix} 1 & 2 & 3 \ 4 & 5 & 6 \end{pmatrix} ] To find the transpose of A, interchange the rows and columns: [ A^T = \begin{pmatrix} 1 & 4 \ 2 & 5 \ 3 & 6 \end{pmatrix} ]

Properties of Transpose

  1. The transpose of a transpose matrix is the original matrix.
  1. Transpose of a sum of matrices is equal to the sum of their transposes.
  1. Transpose of a product of matrices is equal to the product of their transposes taken in reverse order.
  1. Scalar multiplication can be done before or after the transpose operation.

Property 1: Transpose of a Transpose

If A is a matrix, then (A^T)^T = A.

Property 2: Transpose of a Sum

If A and B are matrices of the same size, then (A + B)^T = A^T + B^T. Example: If A = [ 1 2 ] [ 3 4 ] and B = [ 5 6 ] [ 7 8 ] then (A + B)^T = [ 6 8 ] [ 10 12 ]

and A^T + B^T = [ 1 3 ] [ 2 4 ] + [ 5 7 ] [ 6 8 ] = [ 6 10 ] [ 8 12 ]

Property 3: Transpose of a Product

If A and B are matrices such that AB is defined, then (AB)^T = B^T * A^T. Example: If A = [ 1 2 ] [ 3 4 ] and B = [ 5 6 ] [ 7 8 ] then (AB)^T = [ 19 22 ] [ 43 50 ]

and B^T * A^T = [ 5 7 ] [ 6 8 ] * [ 1 3 ] [ 2 4 ] = [ 19 22 ] [ 43 50 ]

Property 4: Scalar Multiplication

If A is a matrix and k is a scalar, then (kA)^T = k(A^T). Example: If A = [ 1 2 ] [ 3 4 ] and k = 2 then (2A)^T = [ 2 4 ] [ 6 8 ]

and k(A^T) = 2 [ 1 3 ] [ 2 4 ] = [ 2 4 ] [ 6 8 ]

Properties Summary

  1. (A^T)^T = A
  1. (A + B)^T = A^T + B^T
  1. (AB)^T = B^T * A^T
  1. (kA)^T = k(A^T)

Transpose of an n x m matrix

  • The transpose of an n x m matrix changes it into an m x n matrix. Example: If A = [ 1 2 3 ] [ 4 5 6 ] then A^T = [ 1 4 ] [ 2 5 ] [ 3 6 ]

Properties of Transpose (contd.)

  1. The transpose of the sum of three matrices is equal to the sum of their transposes.
  1. The transpose of the product of three matrices is equal to the product of their transposes taken in reverse order.
  1. The product of a matrix and its transpose is a symmetric matrix. End of Slide 10

Properties of Transpose (contd.)

Property 5: Transpose of the Sum of Three Matrices

If A, B, and C are matrices of the same size, then (A + B + C)^T = A^T + B^T + C^T.

Property 6: Transpose of the Product of Three Matrices

If A, B, and C are matrices such that ABC is defined, then (ABC)^T = C^T * B^T * A^T.

Property 7: Product of a Matrix and its Transpose

If A is a matrix of size n x m, then A * A^T is a symmetric matrix of size n x n.

Determinant of a Matrix

  • The determinant of a square matrix A is a scalar value that represents certain properties of the matrix.
  • The determinant is denoted by det(A) or |A|.

Properties of Determinants

  1. The determinant of a 2 x 2 matrix [a b; c d] is given by det([a b; c d]) = ad - bc.
  1. The determinant of a 3 x 3 matrix A can be calculated using the cofactor expansion method.
  1. If A is an invertible matrix, then det(A) ≠ 0. Otherwise, if A is non-invertible, then det(A) = 0.

Cofactor Expansion Method for 3 x 3 Matrix

For a 3 x 3 matrix A = [a b c; d e f; g h i], the determinant det(A) is given by: det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)

Example: Determinant Calculation

Let A = [2 1 3; 4 -2 5; 3 0 1] Using the cofactor expansion method: det(A) = 2(1 * 1 - 0 * 5) - 1(4 * 1 - 3 * 5) + 3(4 * 0 - -2 * 1) = 2(1) - 1(7) + 3(-2) = 2 - 7 - 6 = -11 Therefore, det(A) = -11.

Properties of Determinants (contd.)

  1. The determinant of a product of matrices is equal to the product of their determinants.
  1. The determinant of the transpose of a matrix is equal to the determinant of the original matrix.
  1. The determinant of an upper triangular matrix is equal to the product of its diagonal elements.
  1. The determinant of a lower triangular matrix is equal to the product of its diagonal elements.

Property 4: Determinant of a Product

If A and B are square matrices of the same order, then det(A * B) = det(A) * det(B).

Property 5: Determinant of the Transpose

If A is a square matrix, then det(A^T) = det(A).

Property 6: Determinant of an Upper Triangular Matrix

If A is an upper triangular matrix, then det(A) = a₁₁ * a₂₂ * … * aₙₙ, where aᵢⱼ is the entry in the i-th row and j-th column.

Property 7: Determinant of a Lower Triangular Matrix

If A is a lower triangular matrix, then det(A) = a₁₁ * a₂₂ * … * aₙₙ, where aᵢⱼ is the entry in the i-th row and j-th column.

Determinant of any Singular Matrix

  • A matrix A is said to be singular if det(A) = 0.
  • If A is singular, then it does not have an inverse.

Solving Systems of Linear Equations using Determinants

  • Determinants can be used to solve systems of linear equations.
  • The determinants of coefficient matrix and augmented matrix can help determine the existence and uniqueness of solutions.

Cramer’s Rule

  • Cramer’s Rule provides a solution for systems of linear equations using determinants.
  • It involves finding the determinants of sub-matrices formed by replacing each column of the coefficient matrix with the column of constants.

Cramer’s Rule (contd.)

Given a system of linear equations: a₁₁x + a₁₂y + a₁₃z = b₁ a₂₁x + a₂₂y + a₂₃z = b₂ a₃₁x + a₃₂y + a₃₃z = b₃

  • The solution for x can be found using the determinant: x = det( [b₁ a₁₂ a₁₃] ) / det( [a₁₁ a₁₂ a₁₃] )
  • Similarly, the solutions for y and z can be found using the determinants of the appropriate sub-matrices.

Example: Solving a System of Linear Equations using Cramer’s Rule

Given the system of equations:

2x + 3y - z = 5 x - 2y + 4z = -7

3x + y + z = 9 To find the values of x, y, and z:

  • Calculate the determinants for x, y, and z using Cramer’s Rule.
  • Evaluate the determinants and divide by the determinant of the coefficient matrix to find the values of x, y, and z. End of Slide 20.

Determinant of a Matrix (contd.)

  1. If two rows (or columns) of a matrix are interchanged, then the determinant of the new matrix is equal to the negative of the determinant of the original matrix. Example: Let A = [ 1 2 3 ] [ 4 5 6 ] [ 7 8 9 ] If we interchange the first and second rows, the matrix becomes: [ B = [ 4 5 6 \newline 1 2 3 \newline 7 8 9 ] det(B) = -det(A) = -6

Property 9: Determinant of a Diagonal Matrix

The determinant of a diagonal matrix is equal to the product of its diagonal elements. Example: Let A = [ a₁₀ 0 0 ] [ 0 a₂₂ 0 ] [ 0 0 a₃₃ ] det(A) = a₁₀ * a₂₂ * a₃₃

Property 10: Determinant of the Identity Matrix

The determinant of the identity matrix I is equal to 1. Example: Let I = [ 1 0 0 ] [ 0 1 0 ] [ 0 0 1 ] det(I) = 1

Property 11: Determinant of a Scalar Multiple

If A is a matrix and k is a scalar, then det(kA) = k^n * det(A), where n is the order of the matrix A. Example: Let A = [ 2 4 ] [ 6 8 ] and k = 3 det(3A) = (3^2) * det(A) = 9 * det(A)

Matrix Operations and Determinants

  • Matrix addition, subtraction, and scalar multiplication do not affect the determinant of a matrix.
  • The determinant of the sum or difference of two matrices is equal to the sum or difference of their determinants.
  • The determinant of the product of two matrices is equal to the product of their determinants.

Summary of Matrix and Determinant Properties

  1. Transpose of a transpose: (A^T)^T = A
  1. Transpose of a sum: (A + B)^T = A^T + B^T
  1. Transpose of a product: (AB)^T = B^T * A^T
  1. Scalar multiplication: (kA)^T = k(A^T)
  1. Determinant of a product: det(A * B) = det(A) * det(B)
  1. Determinant of transpose: det(A^T) = det(A)
  1. Determinant of upper triangular matrix: det(A) = a₁₁ * a₂₂ * … * aₙₙ
  1. Determinant of interchanged rows/columns: det(B) = -det(A)
  1. Determinant of diagonal matrix: det(A) = a₁₁ * a₂₂ * … * aₙₙ
  1. Determinant of identity matrix: det(I) = 1
  1. Determinant of scalar multiple: det(kA) = k^n * det(A)

Summary (contd.)

  • The transpose of a matrix involves interchanging its rows and columns.
  • The determinant of a matrix is a scalar value that represents certain properties of the matrix.
  • Properties of transpose and determinant help in solving systems of linear equations and performing matrix operations.
  • Cramer’s Rule uses determinants to solve systems of linear equations.
  • Understanding the properties of both transpose and determinant is crucial in various mathematical applications.

Example: Determinant Calculation

Let A = [ 1 2 ] [ 3 4 ] To find the determinant: det(A) = 1(4) - 2(3) = 4 - 6 = -2 Therefore, det(A) = -2.

Example: Solving Systems of Linear Equations

Given the system of equations:

2x + 3y - z = 5 x - 2y + 4z = -7

3x + y + z = 9 Using Cramer’s Rule:

  • Evaluate the determinants for x, y, and z by substituting the constant column with the column of constants.
  • Divide each determinant by the determinant of the coefficient matrix to find the values of x, y, and z. Example: x = det([ 5 3 -1; -7 -2 4; 9 1 1 ]) / det([ 2 3 -1; 1 -2 4; 3 1 1 ])

Conclusion

  • Understanding the properties of matrix transpose and determinant is essential in various mathematical applications, including solving systems of linear equations.
  • The transpose of a matrix involves interchanging rows and columns, while the determinant represents certain properties of a matrix.
  • Utilizing the properties of transpose and determinant can simplify calculations and provide valuable insights in mathematics.