Match List - I with List - II.

Choose the correct answer from the options given below : 

(A) - (III), (B) - (I), (C) - (II), (D) - (IV)

(A) A is a square matrix of order 3 and |2A| = k|A|, then k is

|2A| = k|A|

$2^3|A|=k|A|⇒k=8$

(B) Value of \(\begin{vmatrix}1 & a & b+c\\1 & b & c+a\\1 & c & a+b\end{vmatrix}\)

We know that for a 3×3 matrix $A=\begin{bmatrix}​a&f&c\\d&e&f\\g&h&i​\end{bmatrix}$, the determinant is:

$∣A∣=a(ei−fh)−b(di−fg)+c(dh−eg)$

given matrix is $A=\begin{bmatrix}​a&f&c\\d&e&f\\g&h&i​\end{bmatrix}$

then det. of A is ⇒ $∣A∣=1[b(a+b)−c(c+a)]−a[1(a+b)−1(c+a)]+(b+c)[1(c)−1(b)]$

$=1(ab+b^2−c^2−ac)−a(a+b−c−a)+(b+c)(c−b)$

$=ab+b^2−c^2−ac−a^2−ab+ac+a^2+bc−b^2+c^2−bc$

$=ab−ab+b^2−b^2−c^2+c^2−ac+ac−a^2+a^2+bc−bc$

$=0$

(C) Matrix\(\begin{bmatrix}5-x & x+1 \\2 & 4\end{bmatrix}\) is singular, then x =

\(\begin{bmatrix}5-x & x+1 \\2 & 4\end{bmatrix}=0\)

$⇒20−4x−2x−2=0$

$⇒18−6x=0$

$⇒6x=18$

$⇒x=\frac{18}{6}​=3$

(D) If \(A = (a_{ij})=\begin{bmatrix}5 & 3 & 8\\2 & 0 & 1\\1 & 2 & 3\end{bmatrix}\) then the minor of the element a₂₃ is

We need the minor of element a₂₃​, the row II and column III would get cut.

The determinant of the remaining square matrix, i.e. det$\begin{bmatrix}5&13\\1&2\end{bmatrix}$ is the required minor.

∴ minor of a₂₃​ = 5 × 2 − 1 × 3 = 10 − 3 = 7.

So, correct matching is 

(A) - (III), (B) - (I), (C) - (II), (D) - (IV)

Option 1 is correct.