TIFR CSE 2021 | Part B | Question: 8

Question

Let $A$ and $B$ be two matrices of size $n \times n$ and with real-valued entries. Consider the following statements.

1. If $AB = B$, then $A$ must be the identity matrix.
2. If $A$ is an idempotent (i.e. $A^{2} = A$) nonsingular matrix, then $A$ must be the identity matrix.
3. If $A^{-1} = A$, then $A$ must be the identity matrix.

Which of the above statements $\text{MUST}$ be true of $A$?

• A. $1, 2 $ and $3$
• B. Only $2$ and $3$
• C. Only $1$ and $2$
• D. Only $1$ and $3$
• E. Only $2$

Answer 1

(1). If $AB =B$, then A must be the identity matrix. $\qquad \rightarrow$ False

If $B$ is invertible, then $B^{-1}$ exists, 

and $ABB^{-1} = BB^{-1} \implies AI = I \implies A=I$

But $B$ may not be invertible, in such case $A$ may or may not be identity matrix.

For example, If $A=B$, then we have $A^2 = A$, here $A$ can be a non-singular matrix.

For $A = B = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}$, we have $AB =B$, and $A$ is not identity.

Hence this is False.

 

(2). If $A$ is an idempotent nonsingular matrix, then $A$ must be the identity matrix. $\qquad \rightarrow$ True.

$A$ is nonsingular, means $A^{-1}$ exists,

$A^2 = A \implies AAA^{-1} = AA^{-1}  \implies AI = I \implies A=I$

Hence true.

 

(3).  If $A^{−1}=A$, then $A$ must be the identity matrix. $\qquad \rightarrow$ False.

$A$ is an Involutory matrix,  It not necessary that $A=I$

For example: $P = P^{-1} = \begin{bmatrix} 1 & 0 & 0\\ 0 & 0 &1 \\ 0&1&0 \end{bmatrix}$ $\qquad$ (Interchange row/column 2 and 3)

Hence False.

Option (E) is correct.

Answer 2

1.) Here, B can also be a zero matrix, then also it's possible to form AB = B.

No need to solve further as all other options except E contains 1.
So, the correct answer is indeed E.
2.) Yes, that's TRUE, as we know that an idempotent matrix will only have the eigen values as 0 or 1...
So, for all the eigen values to be 1, necessary to make it an identity matrix, we need to simplify the characteristic equation : (x-1)^n = 0..
And, solving this equation x=1, being the eigen value of matrix, will give A to be identity matrix.. Other than that there's not a single possible way to keep A as an idempotent but not Identity matrix...

3.) This is also false, as there are many matrices for which the matrix is equal to the inverse, but, they aren't the Identity...
{E.g.: Change the rows of the identity matrix}

Hence, E is correct {Only 2}