Matrix Transpose

This lesson will cover matrix transposition, which is a fundamental operation in linear algebra. The transpose of a matrix is obtained by flipping the matrix over its diagonal. This operation is denoted by the superscript “T” or by the prime

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Given a matrix $m\times n$, find the transpose:

Transpose $A^T$ is the $n\times m$

• $1^{st}$ i

Example

$${\left[\begin{array}{cccccc}1& 9& 8& 3& 4& 4\\ 2& 1& 0& 4& 0& 1\\ 3& 7& 8& 7& 1& 1\end{array}\right]}^T$$

The transpose of this matrix is:

$$=\left[\begin{array}{ccc}1& 2& 3\\ 9& 1& 7\\ 8& 0& 8\\ 3& 4& 7\\ 4& 0& 1\\ 4& 1& 1\end{array}\right]$$

Remember, the transpose is not a rotation, it is a reflection.

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Definition

Given a square $n\times n$ matrix $A$, the matrix inverse $A^{-1}$, when it exists, is the one and only matrix such that:

$$A{A}^{-1}=I_{n\times n}=A^{-1}A$$

$$[A\mid I]{\to }^{RREF}[I\mid A^{-1}]$$