Which of the following is not the property of transpose of a matrix?

The correct option is (AB)’=(BA)’.

Let us understand the transpose of a matrix and analyze each property step-by-step to see why this option is not a property of the transpose of a matrix.

The transpose of a matrix $A$, denoted by $A^{\prime }$ or $A^T$, is obtained by swapping its rows and columns.

Let us evaluate each of the given options:

1. Double Transpose Property: $(A^{\prime }{)}^{\prime }=A$
Taking the transpose of a matrix swaps its rows and columns. If we transpose it again, the rows and columns are swapped back to their original positions. Therefore, $(A^{\prime }{)}^{\prime }=A$ is a valid property.

2. Sum Property: $(A+B{)}^{\prime }=A^{\prime }+B^{\prime }$
The transpose of the sum of two matrices is equal to the sum of their individual transposes. This is a standard linear property of matrix transposition, meaning $(A+B{)}^{\prime }=A^{\prime }+B^{\prime }$ is also a valid property.

3. Scalar Multiplication Property: $(kA{)}^{\prime }=k{A}^{\prime }$
Multiplying a matrix by a scalar $k$ scales all its elements. Transposing this scaled matrix yields the same result as transposing the matrix first and then multiplying by $k$. Note that the option lists "KA’" (using capital K), which is a typographical variant of $k{A}^{\prime }$, representing the same valid property.

4. Product Property (Reversal Law): $(AB{)}^{\prime }=B^{\prime }{A}^{\prime }$
According to the reversal law of matrix transpose, the transpose of the product of two matrices $A$ and $B$ is equal to the product of their transposes in reverse order:
$(AB{)}^{\prime }=B^{\prime }{A}^{\prime }$
The option statement asserts that $(AB{)}^{\prime }=(BA{)}^{\prime }$. In general, matrix multiplication is not commutative ($AB\ne BA$), which means that $(AB{)}^{\prime }$ is not equal to $(BA{)}^{\prime }$ for arbitrary matrices. Thus, this is not a property of the transpose of a matrix.