ISRO 2014- Matrices [mech]

Question

Let A=  $\begin{bmatrix} 0 &0 & -1\\ 0& -1 & 0\\ -1 &0 & 0 \end{bmatrix}$

The only correct statement about the matrix A is

(a) A is a ZERO matrix

(b) A2=I

(c) A-1 does not exist

(d) A= (-1) I, where I is a Unit matrix

Answer 1

Here the answer is b) i.e. A^{2 =} I 

let's see each options,

A) A is a zero matrix - FALSE

Zero matrices are those matrices which when multiplied to any other matrix generate a zero matrix.

ref :-https://youtu.be/LOf8bfjiLow

B) A(square) = Identity matix - TRUE

simply multiply the given matrix to itself you will get identity matrix.

C) A(inverse) does not exist - FALSE

Generally A(inverse) = (Adjoint of A) / determinant(A)

here the determinant is 1 which is not equal to zero  

hence A(inverse) is exist.

D) A= (-1) I, where I is a Unit matrix - FAlSE

Even though it is looking like the identity matrix which is written after applying some row interchange operations. But it is not the Identity matix because identity matrix has the property that when ever we multiply any matrix let's say X then the result will be matrix X itself (as it is).

But if we multiply it with any other matrix then the resultant matrix is not as it is which it was initially, there are some rows which interchange itself in the resultant matrix.

Hence B) is correct answer.

Answer 2

We can check by eliminating the Options...

See..
(A) A is a zero matrix.... Obviously not, as all elements aren't zero..!!
(C) A^(-1) doesn't exist is false, as by checking for the 3rd elemnt of 1st row, we'll get it as 1...!!
(D) A = -I. This is also false, as I has all principal diagonal elements 1 and others zero, but here, anti-diagonal elements are -1 and others 0.

Hence, we can easily conclude that (B) which says A^2 is identity matrix is TRUE...!!