Q81 / 180 · COMEDK 2025 (10 May Shift 1) · Mathematics
The sum of three numbers is 6 .
Twice the third number, when added to the first number gives 7 ,
On adding the sum of the second and third numbers to thrice the first number, we get 12 .
The above situation can be represented in matrix form as $A X=B$.
Then the $|\operatorname{adj} A|$ is equal to
- Correct answerA$16$
- B$-64$
- C$-4$
- D$4$
Solution
Let the three numbers be $x, y, z$.
The given conditions are:
1. $x + y + z = 6$
2. $x + 0y + 2z = 7$
3. $3x + y + z = 12$
The system of equations in matrix form $AX = B$ is:
$\begin{pmatrix} 1 & 1 & 1 \\ 1 & 0 & 2 \\ 3 & 1 & 1 \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 6 \\ 7 \\ 12 \end{pmatrix}$
Calculate the determinant of matrix $A$:
$|A| = 1(0 - 2) - 1(1 - 6) + 1(1 - 0)$
$|A| = -2 + 5 + 1 = 4$
The property of the adjoint matrix is $|\operatorname{adj} A| = |A|^{n-1}$, where $n$ is the order of the matrix.
Here $n = 3$, so $|\operatorname{adj} A| = |A|^{3-1} = |A|^2$.
$|\operatorname{adj} A| = 4^2 = 16$.
Answer: $16$
The given conditions are:
1. $x + y + z = 6$
2. $x + 0y + 2z = 7$
3. $3x + y + z = 12$
The system of equations in matrix form $AX = B$ is:
$\begin{pmatrix} 1 & 1 & 1 \\ 1 & 0 & 2 \\ 3 & 1 & 1 \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 6 \\ 7 \\ 12 \end{pmatrix}$
Calculate the determinant of matrix $A$:
$|A| = 1(0 - 2) - 1(1 - 6) + 1(1 - 0)$
$|A| = -2 + 5 + 1 = 4$
The property of the adjoint matrix is $|\operatorname{adj} A| = |A|^{n-1}$, where $n$ is the order of the matrix.
Here $n = 3$, so $|\operatorname{adj} A| = |A|^{3-1} = |A|^2$.
$|\operatorname{adj} A| = 4^2 = 16$.
Answer: $16$
Hence, the correct answer is option: 'A'.