Properties of Transpose of the Matrices

Concept of Transpose

The transpose of a matrix is a fundamental operation that swaps its rows and columns. If a matrix $A$ has entry $a_{ij}$ in the $i$-th row and $j$-th column, then its transpose, denoted $A'$ (or $A^T$), has the entry $a_{ji}$ in the $i$-th row and $j$-th column. This operation has several predictable and useful properties, stated below, which can be verified by taking suitable examples of matrices of appropriate orders.

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Properties of Transpose

The following four properties hold for any matrices $A$ and $B$ of suitable orders (so that addition and multiplication are defined), and for any constant $k$.

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Property (I): Double Transpose

Statement: The transpose of the transpose of a matrix is the original matrix itself.

Explanation: If $A = [a_{ij}]$, then $A' = [a_{ji}]$. Taking the transpose of $A'$ swaps the rows and columns again, returning the entry $a_{ji}$ to its original position $a_{ij}$. Thus the original matrix is recovered.

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Property (II): Transpose of a Scalar Multiple

Statement: For any matrix $A$ and constant $k$, the transpose of $kA$ equals $k$ times the transpose of $A$.

Explanation: Multiplying by $k$ scales every entry by $k$; transposing then swaps rows and columns, with each entry still multiplied by $k$. Taking the transpose first and multiplying by $k$ afterwards gives the same result, because scalar multiplication commutes with transposition.

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Property (III): Transpose of a Sum

Statement: For any two matrices $A$ and $B$ of the same order, the transpose of their sum equals the sum of their transposes.

Explanation: Matrix addition is entry-wise: adding $A$ and $B$ forms each entry $a_{ij} + b_{ij}$, which the transpose places at position $(j,i)$. On the other hand, $A'$ and $B'$ have entries $a_{ji}$ and $b_{ji}$, whose sum $a_{ji} + b_{ji}$ sits at the same position. The two results are identical.

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Property (IV): Transpose of a Product

Statement: For any two matrices $A$ and $B$ such that $AB$ is defined, the transpose of the product equals the product of their transposes in reverse order.

$(AB)' = B'A'$ …