# Matrix transpose

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Definition.

**Transposing the matrix**is an operation on the matrix in which its rows and columns are swapped:

a^{T}_{ij} = a_{ji}

## Properties of transpose matrix

- If matrix A has a size of n×m, then the transposed matrix A
^{T}has a size of m×n; - (A
^{T})^{T}= A; - (k · A)
^{T}= k · A^{T}; - (A + B)
^{T}= A^{T}+ B^{T}; - (A · B)
^{T}= B^{T}· A^{T}.

## Examples of matrix transpose

Example 1.

Find the transposed matrix A^{T}for matrix

A = | 4 | 2 | . | ||

9 | 0 |

**Solution:**

A^{T} = |
4 | 9 | ||

2 | 0 |

Example 2

Find the transposed matrix A^{T}for matrix

A = | 2 | 1 | . | ||

-3 | 0 | ||||

4 | -1 |

**Solution:**

A^{T} = |
2 | -3 | 4 | ||

1 | 0 | -1 |

Example 3

Find the transposed matrix A^{T}for matrix

A = | 2 | -3 | 4 | . | ||

1 | 0 | -1 |

**Solution:**

A^{T} = |
2 | 1 | ||

-3 | 0 | |||

4 | -1 |

MatrixMatrix Definition. Main informationSystem of linear equations - matrix formTypes of matricesMatrix scalar multiplicationAddition and subtraction of matricesMatrix multiplicationTranspose matrixElementary matrix operationsDeterminant of a matrixMinors and cofactors of a matrixInverse matrixLinearly dependent and independent rowsRank of a matrix

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