Question

If A is an m × n matrix and B is a matrix such that both AB and BA are 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.

It is given that, A is an m × n matrix.

Let the order of matrix B be p × q.

For AB to be defined,

n = p .....(1) (Number of columns of A is same as the number of rows of B)

For BA to be defined,

q = m .....(2) (Number of columns of B is same as the number of rows of A)

From (1) and (2), we conclude that the order of matrix B be n × m.

If A is an m × n matrix and B is a matrix such that both AB and BA are defined, then the order of B is ___n × m___.