If $A$ and $B$ are matrices of order $m \times n$ and $n \times m$ respectively, then order of matrix $B^T(A^T)^T$ is -
Step-by-Step Solution
Key Concept: For a matrix product $XY$ to be defined, the number of columns in the first matrix must equal the number of rows in the second, utilizing the properties $(A^T)^T = A$ and that the order of $B^T$ is $m imes n$ if $B$ is $n imes m$.
Order of $A = m \times n \Rightarrow$ Order of $(A^T)^T = A = m \times n$.<br>Order of $B = n \times m \Rightarrow$ Order of $B^T = m \times n$.<br>For the product $B^T(A^T)^T$ to be defined, the number of columns of $B^T$ must equal the number of rows of $(A^T)^T$.<br>Number of columns of $B^T = n$.<br>Number of rows of $(A^T)^T = m$.<br>Since $n$ is not necessarily equal to $m$, the product is not defined in general. Ans. (D)
Correct Answer: D