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Determinants — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDeterminants1 Mark
MCQ
If $A =\left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right]$, then value of $A ^{-1}$ :
  • A
    $\left[\begin{array}{ll}4 & 2 \\ 3 & 1\end{array}\right]$
  • ✓
    $\left[\begin{array}{cc}-2 & 1 \\ \frac{3}{2} & -1 / 2\end{array}\right]$
  • C
    $\left[\begin{array}{cc}4 & -3 \\ 2 & 1\end{array}\right]$
  • D
    $\left[\begin{array}{ll}-4 & 3 \\ -2 & 1\end{array}\right]$
✓

Answer

Correct option: B.
$\left[\begin{array}{cc}-2 & 1 \\ \frac{3}{2} & -1 / 2\end{array}\right]$
(B)
$A_{11}=4, A_{12}=-3, A_{21}=-2, A_{22}=1$$
\operatorname{adj} A=\left[\begin{array}{cc}
4 & -3 \\
-2 & 1
\end{array}\right]^{\prime}=\left[\begin{array}{cc}
4 & -2 \\
-3 & 1
\end{array}\right]
$
$
\begin{aligned}
|A| & =4-6=-2 \neq 0 \\
A^{-1}=\frac{\operatorname{adj} A}{|A|} & =\frac{-1}{2}\left[\begin{array}{cc}
4 & -2 \\
-3 & 1
\end{array}\right]=\left[\begin{array}{cc}
-2 & 1 \\
\frac{3}{2} & -1 / 2
\end{array}\right]
\end{aligned}
$
Hence correct option is (B).

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Let $n_1$ and $n_2$ be the number of red and black balls, respectively, in box $I$. Let $n_3$ and $n_4$ be the number of red and black balls, respectively, in box $II$.

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