Matrices — Elementary Row Operations
DAILY_CHALLENGE
Grade None
Question:
Which one of the following matrices can be obtained by performing elementary row transformations on the $3 \times 3$ identity matrix?
$\begin{bmatrix}1&1&1\\1&1&1\\1&1&1\end{bmatrix}$
$\begin{bmatrix}1&1&1\\2&3&4\\1&2&1\end{bmatrix}$
$\begin{bmatrix}1&1&1\\2&3&4\\2&5&8\end{bmatrix}$
$\begin{bmatrix}1&1&1\\-1&1&2\\0&2&3\end{bmatrix}$
Step-by-Step Solution
Key Concept: Elementary row operations are equivalent to left-multiplication by elementary matrices, each invertible. Any product of invertible matrices is invertible, so the result always has non-zero determinant.
**Step 1: Key invariant of elementary row operations**
Elementary row operations on $I$ produce only invertible matrices (determinant $\ne 0$). So we eliminate any option with $\det = 0$.
**Step 2: Check each determinant**
(A) All rows identical $\Rightarrow \det = 0$. ✗
(B) $\det = 1(3\cdot1-4\cdot2)-1(2\cdot1-4\cdot1)+1(2\cdot2-3\cdot1) = (3-8)-(2-4)+(4-3) = -5+2+1 = -2 \ne 0$. ✓
(C) $\det = 1(24-20)-1(16-8)+1(10-6) = 4-8+4 = 0$. ✗
(D) $\det = 1(3-4)-1(-3-0)+1(-2-0) = -1+3-2 = 0$. ✗
**Step 3: Verify (B) is reachable from $I$**
Since $\det(B) = -2 \ne 0$, matrix $B$ is invertible and hence expressible as a product of elementary matrices, confirming it can be obtained from $I$ by row operations.
Correct Answer: B