Transpose rules for matrix products, sums, and inverses

Taking a transpose swaps rows and columns. We can apply that idea to a whole matrix expression, too. Let’s see what happens to a product, a sum, and an inverse.

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Reverse the factors when transposing a product

$(XY{)}^T=Y^T{X}^T$

Use the following two matrices.

$X=\left[\begin{array}{rr}2& 1\\ -3& 0\end{array}\right]$

$Y=\left[\begin{array}{rr}-6& 0\\ 2& -1\end{array}\right]$

$XY=\left[\begin{array}{rr}-10& -1\\ 18& 0\end{array}\right]$

Swapping the rows and columns of this product gives the same result as multiplying the two transposes in reverse order.

$Y^T{X}^T=\left[\begin{array}{rr}-10& 18\\ -1& 0\end{array}\right]$

Transpose the terms in a sum

$(X+Y{)}^T=X^T+Y^T$

Here $X+Y$ has rows $(-4,1)$ and $(-1,-1)$. Its transpose has rows $(-4,-1)$ and $(1,-1)$. Adding $X^T$ and $Y^T$ gives those same rows.

Transpose an inverse when the matrix is invertible

$(X^{-1}{)}^T=(X^T{)}^{-1}$

Transposing $X{X}^{-1}=I$ gives $(X^{-1}{)}^T{X}^T=I$. Transposing the product in the other order gives $X^T(X^{-1}{)}^T=I$. Together, they show that $(X^{-1}{)}^T$ is the inverse of $X^T$.

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