Question

 If \[A, B, C\] are three \[n \times n\] matrices, then find the equivalent value of \[{\left( {ABC} \right)^\prime }\].
A. \[A'B'C'\]
B. \[C'B'A'\]
C. \[B'C'A'\]
D. \[B'A'C'\]

Answer 1

Hint:
Here, three \[n \times n\] matrices are given. First, consider the product of the first two matrices as a single matrix. Then, apply the multiplication property of the transpose of the matrices and solve it. Again, apply the multiplication property of transpose of the matrices on the product of the matrices. In the end, simplify the transpose and calculate the equivalent value of the required problem.



Formula Used:
Transpose property of product of the matrices: \[{\left( {AB} \right)^\prime } = B'A'\]




Complete step-by-step answer:
Given that, \[A, B\] and \[ C\] are \[n \times n\] matrices.

Let’s calculate the value of \[{\left( {ABC} \right)^\prime }\].
Consider the product of the matrices \[A, B\] as a one matrix and apply the transpose of the product of the matrices.
Then,
\[{\left( {ABC} \right)^\prime } = {\left( {\left( {AB} \right)C} \right)^\prime }\]
Apply the multiplication property of the transpose of the matrices on the right-hand side.
\[{\left( {ABC} \right)^\prime } = C'{\left( {AB} \right)^\prime }\]
Again, apply the multiplication property of the transpose of the matrices on the right-hand side.
We get,
\[{\left( {ABC} \right)^\prime } = C'B'A'\]
Hence the correct option is B.

Additional information
Matrix: Matrix is a rectangular array where the elements are arranged along rows and columns. If the number of rows and columns of the matrix is the same, then the matrix is known as a square matrix.



Note:
The multiplication property of transpose is that the transpose of a product of the matrices will be equal to the product of the transpose of individual matrices in reverse order.