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# If $A$ is $3 \times 4$ matrix and ${B^T}$ is matrix such that ${A^T}B$ and $B{A^T}$ are both defined, then $B$ is of the type.A) $3 \times 4$B) $3 \times 3$C) $4 \times 3$D) $4 \times 4$

Last updated date: 13th Jun 2024
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Hint: 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.

We have been given that the matrix $A$ is $3 \times 4$.
So, this implies ${A^T}$ is $4 \times 3$.
Here number of columns of ${A^T}$= 3
Here number of rows of ${A^T}$= 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 $P{\text{ }} \times {\text{ }}Q$ with P be number of rows and Q is number of columns
Since ${A^T}B$ is defined, so number of columns of ${A^T}$ must be equal to number of rows of B,
therefore, P = 3.
Also, $B{A^T}$ is defined, so the number of columns of B must be equal to number of rows of ${A^T}$,
then Q = 4.
Therefore, matrix B is $3 \times 4$.
So, the order of the matrix ${B^T}$ is $4 \times 3$.

So, option (C) is the correct answer.

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