Match List-I with List-II
| List-I | List-II | ||
|---|---|---|---|
| Matrix | Real value(s) of k for which the matrix is singular | ||
| (A) | \(\begin{bmatrix}2 & k & 1\\ 1 & 3 & 0\\ 0 & 1 & 1\end{bmatrix}\) | (I) | \(k = \pm 2\) |
| (B) | \(\begin{bmatrix}1 & 2 & 3\\ 2 & k & 6\\ 0 & 1 & 1\end{bmatrix}\) | (II) | \(k = 7\) |
| (C) | \(\begin{bmatrix}k & 1 & 0\\ 4 & k & 0\\ 1 & 2 & 3\end{bmatrix}\) | (III) | No real value of k |
| (D) | \(\begin{bmatrix}k & -1 & 0\\ 4 & k & 0\\ 5 & 6 & 1\end{bmatrix}\) | (IV) | \(k = 4\) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A)-(II), (B)-(IV), (C)-(I), (D)-(III)
A matrix is singular when its determinant is zero.
(A) \(\begin{bmatrix}2 & k & 1\\ 1 & 3 & 0\\ 0 & 1 & 1\end{bmatrix}\)
This matches (II) \(k = 7\).
\(2(3 - 0) - k(1 - 0) + 1(1 - 0) = 7 - k\)
\(7 - k = 0\)
\(k = 7\)
(B) \(\begin{bmatrix}1 & 2 & 3\\ 2 & k & 6\\ 0 & 1 & 1\end{bmatrix}\)
This matches (IV) \(k = 4\).
\(1(k - 6) - 2(2 - 0) + 3(2 - 0) = k - 4\)
\(k - 4 = 0\)
\(k = 4\)
(C) \(\begin{bmatrix}k & 1 & 0\\ 4 & k & 0\\ 1 & 2 & 3\end{bmatrix}\)
This matches (I) \(k = \pm 2\).
Expanding along the third column, only the entry 3 is non-zero:
\(3(k^{2} - 4) = 0\)
\(k = \pm 2\)
(D) \(\begin{bmatrix}k & -1 & 0\\ 4 & k & 0\\ 5 & 6 & 1\end{bmatrix}\)
This matches (III) No real value of k.
Expanding along the third column:
\(1\big(k^{2} - (-1)(4)\big) = k^{2} + 4\)
Since \(k^{2} + 4 \ge 4\) for every real k, the determinant is never zero.