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Question: If \[A\] is \[3 \times 4\] matrix and \[{B^T}\] is matrix such that \[{A^T}B\] and \[B{A^T}\] are bo...

If AA is 3×43 \times 4 matrix and BT{B^T} is matrix such that ATB{A^T}B and BATB{A^T} are both defined, then BB is of the type.
A) 3×43 \times 4
B) 3×33 \times 3
C) 4×34 \times 3
D) 4×44 \times 4

Explanation

Solution

Just as with adding matrices, the sizes of the matrices matter when we are multiplying. For matrix multiplication to work, the columns of the second matrix have to have the same number of entries as do the rows of the first matrix.
The transpose of a certain matrix is its number of rows are interchanged to number of columns and number of columns are interchanged to number of rows.
We are going to use the above two concepts to solve the problem.

Complete step-by-step answer:
We have been given that the matrix AA is 3×43 \times 4.
So, this implies AT{A^T} is 4×34 \times 3.
Here number of columns of AT{A^T}= 3
Here number of rows of AT{A^T}= 4
Now for matrix product AB between matrices A and B is defined only if the number of columns in A equals the number of rows in B
We will assume, the B matrix be P × QP{\text{ }} \times {\text{ }}Q with P be number of rows and Q is number of columns
Since ATB{A^T}B is defined, so number of columns of AT{A^T} must be equal to number of rows of B,
therefore, P = 3.
Also, BATB{A^T} is defined, so the number of columns of B must be equal to number of rows of AT{A^T},
then Q = 4.
Therefore, matrix B is 3×43 \times 4.
So, the order of the matrix BT{B^T} is 4×34 \times 3.

So, option (C) is the correct answer.

Note: When you are noting number of columns and rows of the matrix AT{A^T} then you should take into consideration the number rows and columns of the matrix AT{A^T} and not the numbers of rows and columns of the matrix A.