Matrices - Transpose of Matrix

  • In linear algebra, the transpose of a matrix is an operation that flips the matrix over its diagonal.
  • The transpose of a matrix A is denoted by A^T or A'.

Properties of Transpose

  1. (A^T)^T = A
  1. If A is an m x n matrix, then (A^T)^T = A

Examples

  1. Let A = [1 2 3] be a 1 x 3 matrix. The transpose of A is: A^T = [1] [2] [3]
  1. Let B = [4 5] be a 1 x 2 matrix. The transpose of B is: B^T = [4] [5]
  1. Let C = [6; 7] be a 2 x 1 matrix. The transpose of C is: C^T = [6 7]

Properties of Transpose (contd.)

  1. (A + B)^T = A^T + B^T
  1. (kA)^T = kA^T, where k is a scalar
  1. (AB)^T = B^T A^T, where A and B are matrices

Example

Let A = [1 2] and B = [3 4] be 1 x 2 matrices. The transpose of A + B is: (A + B)^T = A^T + B^T [1 2]^T + [3 4]^T [1] + [3] [2] + [4] [4 6]

Equations Involving Transpose

  • (A^T)^-1 = (A^-1)^T, where A is an invertible matrix
  • (A^T)^-1 = (A^-1)^T, where A is an invertible matrix

Example

Let A = [1 2; 3 4] be a 2 x 2 invertible matrix. The transpose of (A^T)^-1 is: (A^T)^-1 = (A^-1)^T ([1 2; 3 4]^-1)^T ([1.2 -0.5; -1.5 0.5])^T [1.2 -1.5; -0.5 0.5]^T [1.2 -0.5; -1.5 0.5]

Properties of Transpose (contd.)

  • (A^T)^T = A
  • If A is an m x n matrix, then (A^T)^T = A

Examples

  • Let A = [1 2 3] be a 1 x 3 matrix. The transpose of A is: A^T = [1] [2] [3]
  • Let B = [4 5] be a 1 x 2 matrix. The transpose of B is: B^T = [4] [5]
  • Let C = [6; 7] be a 2 x 1 matrix. The transpose of C is: C^T = [6 7]

Properties of Transpose (contd.)

  • (A + B)^T = A^T + B^T
  • (kA)^T = kA^T, where k is a scalar
  • (AB)^T = B^T A^T, where A and B are matrices

Example

  • Let A = [1 2] and B = [3 4] be 1 x 2 matrices. The transpose of A + B is: (A + B)^T = A^T + B^T [1 2]^T + [3 4]^T [1] + [3] [2] + [4] [4 6]

Equations Involving Transpose

  • (A^T)^-1 = (A^-1)^T, where A is an invertible matrix
  • (A^T)^-1 = (A^-1)^T, where A is an invertible matrix

Example

  • Let A = [1 2; 3 4] be a 2 x 2 invertible matrix. The transpose of (A^T)^-1 is: (A^T)^-1 = (A^-1)^T ([1 2; 3 4]^-1)^T ([1.2 -0.5; -1.5 0.5])^T [1.2 -1.5; -0.5 0.5]^T [1.2 -0.5; -1.5 0.5]
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Matrices - Determinant

  • The determinant is a scalar value that is calculated from the elements of a square matrix.
  • It is denoted as det(A) or |A|.
  • Determinants are only available for square matrices.

Properties of Determinant

  1. The determinant of a 2 x 2 matrix A = [a b; c d] can be calculated as ad - bc.
  1. If A is a square matrix, det(A^T) = det(A).
  1. If A is an invertible matrix, det(A^-1) = 1/det(A).

Examples

  1. Let A = [2 3; 4 5] be a 2 x 2 matrix. The determinant of A is: det(A) = (2)(5) - (3)(4) = 10 - 12 = -2
  1. Let B = [1 2 3; 4 5 6; 7 8 9] be a 3 x 3 matrix. The determinant of B is: det(B) = (1)(5)(9) + (2)(6)(7) + (3)(4)(8) - (3)(5)(7) - (2)(4)(9) - (1)(6)(8) = 45 + 84 + 96 - 105 - 72 - 48 = 0

Matrices - Transpose of Matrix (contd.)

  • The transpose of a matrix is a fundamental operation in linear algebra that has several important properties.
  • Let’s explore more properties of transpose.

Properties of Transpose (contd.)

  • (A^T)^T = A (transpose of transpose is the original matrix)
  • If A is an m x n matrix, then (A^T)^T = A (transpose of transpose is the original matrix)

Examples

  • Let A = [1 2 3; 4 5 6] be a 2 x 3 matrix. The transpose of A is: A^T = [1 4] [2 5] [3 6]
  • Let B = [7 8; 9 10; 11 12] be a 3 x 2 matrix. The transpose of B is: B^T = [7 9 11] [8 10 12]

Properties of Transpose (contd.)

  • (A + B)^T = A^T + B^T (transpose of sum is the sum of transposes)
  • (kA)^T = kA^T, where k is a scalar (transpose of scalar matrix multiplication is the scalar times the transpose)
  • (AB)^T = B^T A^T, where A and B are matrices (transpose of product is the product of transposes in reverse order)

Examples

  • Let A = [1 2; 3 4] and B = [5 6; 7 8] be 2 x 2 matrices. The transpose of A + B is: (A + B)^T = A^T + B^T [1 2; 3 4]^T + [5 6; 7 8]^T [1 3; 2 4] + [5 7; 6 8] [6 10; 8 12]
  • Let C = [9 10; 11 12] be a 2 x 2 matrix. The transpose of 3C is: (3C)^T = 3(C^T) 3([9 10; 11 12]^T) 3([9 11; 10 12]) [27 33; 30 36]

Equations Involving Transpose (contd.)

  • (A^T)^-1 = (A^-1)^T, where A is an invertible matrix (transpose of inverse is the inverse of transpose)
  • (A^T)^-1 = (A^-1)^T, where A is an invertible matrix (transpose of inverse is the inverse of transpose)

Examples

  • Let A = [2 1; 4 3] be a 2 x 2 invertible matrix. The transpose of the inverse of A is: (A^T)^-1 = (A^-1)^T ([2 1; 4 3]^-1)^T ([3 -1; -4 2])^T [3 -4; -1 2]^T [3 -1; -4 2]
  • Let B = [5 6 7; 8 9 10; 11 12 13] be a 3 x 3 invertible matrix. The transpose of the inverse of B is: (B^T)^-1 = (B^-1)^T ([5 6 7; 8 9 10; 11 12 13]^-1)^T ([9 -6 5; -18 12 -10; 11 -8 7])^T [9 -18 11; -6 12 -8; 5 -10 7]
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Recap

  • The transpose of a matrix is obtained by interchanging its rows with columns.
  • The transpose has several important properties like being distributive over addition and scalar multiplication.
  • Transpose of a matrix product is the product of the transposes in reverse order.
  • The determinant is a scalar value calculated from the elements of a square matrix.
  • Determinants are only available for square matrices.
  • Determinant has properties like calculating 2x2 matrix determinant, transpose of a determinant is equal to the determinant of the transpose, and the inverse of a determinant is equal to 1 divided by the determinant of the original matrix.

Summary

  • Transposing a matrix is a fundamental operation that flips the matrix over its diagonal.
  • Transpose has properties related to addition, scalar multiplication, and matrix multiplication.
  • Determinants are scalar values computed from square matrices, and they have their own set of properties.
  • Understanding transposition and determinants is crucial for further study of linear algebra and matrix operations.