Question 3

SCQMEDIUM

Which one of the following matrices can be obtained by performing elementary row transformations on the $3 \times 3$ identity matrix?

(A)

$\begin{bmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{bmatrix}$

(B)

$\begin{bmatrix} 1 & 1 & 1 \\ 2 & 3 & 4 \\ 1 & 2 & 1 \end{bmatrix}$

(C)

$\begin{bmatrix} 1 & 1 & 1 \\ 2 & 3 & 4 \\ 2 & 5 & 8 \end{bmatrix}$

(D)

$\begin{bmatrix} 1 & 1 & 1 \\ -1 & 1 & 2 \\ 0 & 2 & 3 \end{bmatrix}$

Detailed Solution

A matrix can be obtained from the identity matrix $I$ using elementary row transformations if and only if it is non-singular (i.e., its determinant is non-zero).

Evaluating the determinants of the given options:

(A) $|A| = 0$ as all rows are identical.

(B) $|B| = 1(3 - 8) - 1(2 - 4) + 1(4 - 3) = -5 + 2 + 1 = -2$ $\neq 0$ .

(D) $|D| = 1(3 - 4) - 1(-3 - 0) + 1(-2 - 0) = -1 + 3 - 2 = 0$ .

Since only matrix (B) is non-singular, it is the only one that can be obtained from the $3 \times 3$ identity matrix via elementary row transformations.

Question 3

Detailed Solution

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