Q66 / 180 · COMEDK 2024 (Evening Shift) · Mathematics
If the matrix $A$ is such that $A\left(\begin{array}{cc}-1 & 2 \\ 3 & 1\end{array}\right)=\left(\begin{array}{cc}-4 & 1 \\ 7 & 7\end{array}\right)$ then $A$ is equal to
- Correct answerA$\left(\begin{array}{cc}1 & -1 \\ 2 & 3\end{array}\right)$
- B$\left(\begin{array}{cc}1 & 1 \\ 2 & -3\end{array}\right)$
- C$\left(\begin{array}{cc}-1 & 1 \\ 2 & 3\end{array}\right)$
- D$\left(\begin{array}{cc}1 & 1 \\ -2 & 3\end{array}\right)$
Solution
Let $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ and $B = \begin{pmatrix} -1 & 2 \\ 3 & 1 \end{pmatrix}$. The given equation is $AB = C$, where $C = \begin{pmatrix} -4 & 1 \\ 7 & 7 \end{pmatrix}$.
To find $A$, we compute $A = CB^{-1}$.
First, find the determinant of $B$: $|B| = (-1)(1) - (2)(3) = -1 - 6 = -7$.
The inverse of $B$ is given by $B^{-1} = \dfrac{1}{|B|} \text{adj}(B) = \dfrac{1}{-7} \begin{pmatrix} 1 & -2 \\ -3 & -1 \end{pmatrix} = \begin{pmatrix} -1/7 & 2/7 \\ 3/7 & 1/7 \end{pmatrix}$.
Now, calculate $A = CB^{-1} = \begin{pmatrix} -4 & 1 \\ 7 & 7 \end{pmatrix} \begin{pmatrix} -1/7 & 2/7 \\ 3/7 & 1/7 \end{pmatrix}$.
The first row of $A$ is: $a_{11} = (-4)(-1/7) + (1)(3/7) = 4/7 + 3/7 = 1$ and $a_{12} = (-4)(2/7) + (1)(1/7) = -8/7 + 1/7 = -1$.
The second row of $A$ is: $a_{21} = (7)(-1/7) + (7)(3/7) = -1 + 3 = 2$ and $a_{22} = (7)(2/7) + (7)(1/7) = 2 + 1 = 3$.
Thus, $A = \begin{pmatrix} 1 & -1 \\ 2 & 3 \end{pmatrix}$.
Answer: $\left(\begin{array}{cc}1 & -1 \\ 2 & 3\end{array}\right)$
To find $A$, we compute $A = CB^{-1}$.
First, find the determinant of $B$: $|B| = (-1)(1) - (2)(3) = -1 - 6 = -7$.
The inverse of $B$ is given by $B^{-1} = \dfrac{1}{|B|} \text{adj}(B) = \dfrac{1}{-7} \begin{pmatrix} 1 & -2 \\ -3 & -1 \end{pmatrix} = \begin{pmatrix} -1/7 & 2/7 \\ 3/7 & 1/7 \end{pmatrix}$.
Now, calculate $A = CB^{-1} = \begin{pmatrix} -4 & 1 \\ 7 & 7 \end{pmatrix} \begin{pmatrix} -1/7 & 2/7 \\ 3/7 & 1/7 \end{pmatrix}$.
The first row of $A$ is: $a_{11} = (-4)(-1/7) + (1)(3/7) = 4/7 + 3/7 = 1$ and $a_{12} = (-4)(2/7) + (1)(1/7) = -8/7 + 1/7 = -1$.
The second row of $A$ is: $a_{21} = (7)(-1/7) + (7)(3/7) = -1 + 3 = 2$ and $a_{22} = (7)(2/7) + (7)(1/7) = 2 + 1 = 3$.
Thus, $A = \begin{pmatrix} 1 & -1 \\ 2 & 3 \end{pmatrix}$.
Answer: $\left(\begin{array}{cc}1 & -1 \\ 2 & 3\end{array}\right)$
Hence, the correct answer is option: 'A'.