Question

If A is 3 × 4 matrix and B is a matrix such that A^TB and BA^T are both defined. Then the order of B is __________.

Answer 1

Let $X={\left[x_{ij}\right]}_{m\times n}$ and $Y={\left[y_{ij}\right]}_{p\times q}$ be two matrices of order m × n and p × q. The multiplication of matrices X and Y is defined if number of columns of X is same as the number of rows of Y i.e. n = p. Also, XY is a matrix of order m × q.

The order of matrix A is 3 × 4. Therefore, the order of matrix A^T is 4 × 3.

Let the order of matrix B be m × n.

For A^TB to be defined,

m = 3 (Number of columns of A^T is same as the number of rows of B)

For BA^T to be defined,

n = 4 (Number of columns of B is same as the number of rows of A^T)

Thus, the order of matrix B is 3 × 4.

If A is 3 × 4 matrix and B is a matrix such that A^TB and BA^T are both defined. Then the order of B is ___3 × 4___.