Let A be a non-singular matrix and B = adj A. Which of the following statements is/are correct?

1. AB = BA

2. AB is a scalar matrix

3. AB can be a null matrix

Select the correct answer using the code given below:

Concept:

1). A square matrix A = [a_ij]_{n x n}, is said to be a scalar matrix if;

     a_ij = 0, when i ≠ j

     a_ij = k, when i = j, for some constant k

2). Non Singular matrix is a square matrix whose determinant is a non-zero value. 

3). Null matrix or zero matrices is a matrix having zero as all its elements.

Formula used:

\(\displaystyle A^{-1}=\frac{adj\ A}{∣\ A ∣ }\) 

Calculation:

Given that, A is a non-singular matrix and B = adj A

We know that

 \(\displaystyle A^{-1}=\frac{adj\ A}{∣\ A ∣ }\)      ------(i)

Statement I: AB = BA

Pre-multiplication by A,

​⇒ \(\displaystyle AA^{-1}=\frac{A.\ adj\ A}{∣\ A ∣ }\)​

⇒ \(\displaystyle I=\frac{A.\ adj\ A}{∣\ A ∣ }\)

⇒ I ∣A∣ = AB      [As B = adj A]      ------(ii)

Post-multiplication by A,

​⇒ \(\displaystyle A^{-1}A=\frac{\ adj\ A.A}{∣\ A ∣ }\)​

⇒ \(\displaystyle I=\frac{\ adj\ A.A}{∣\ A ∣ }\)

⇒ I ∣A∣ = BA      [As B = adj A]      ------(iii)

From equation (ii) and (iii), we have,

AB = BA

Statement I is true.

Statement II: AB is a scalar matrix

From equation (ii), we have, 

AB = ∣A∣ I 

⇒ AB is a scalar matrix

Statement III: AB can be a null matrix

A is a non-singular matrix.

⇒∣A∣ ≠ 0

A (adjA) = ∣A∣ I ≠  0

AB  ≠  0

∴ Only statement I and Statement II are correct.