Consider the following statements in respect of square matrices A and B of same order :

1. If AB is a null matrix, then at least one of A and B is a null matrix.

2. If AB is an identity matrix, then BA = AB.

Which of the above statements is/are correct?

Concept:

1). The null matrix is a matrix having zero as all its elements. The null matrix is also called a zero matrix.

2).  An identity matrix is a given square matrix of any order which contains on its main diagonal elements with value of one

Calculation:

Statement I: If AB is a null matrix, then at least one of A and B is a null matrix.

Let us consider two matrices,

\(A = \begin{bmatrix} 0 &0 \\[0.3em] 0 & 1\\[0.3em] \end{bmatrix}\ and\ B = \begin{bmatrix} 0 &1 \\[0.3em] 0 &0\\[0.3em] \end{bmatrix} \)

AB = \( \begin{bmatrix} 0 &0 \\[0.3em] 0 & 1\\[0.3em] \end{bmatrix}\ \begin{bmatrix} 0 &1 \\[0.3em] 0 &0\\[0.3em] \end{bmatrix}= \begin{bmatrix} 0 &0 \\[0.3em] 0 & 0\\[0.3em] \end{bmatrix}\ \)

Here, both A and B are non-zero matrices. But the product is zero matrices.

⇒ Statement I is incorrect.

Statement II: If AB is an identity matrix, then BA = AB.

AB = I

Pre-multiplying A-1 on both sides.

⇒ A^-1AB = A^-1I      [Considering A is invertible]

⇒ B = A^-1

Post-multiplying A on both sides.

⇒ BA = A^-1A

⇒ BA = I

⇒ Statement II is correct.

∴ Only statement II is correct.