Q113 / 180 · COMEDK 2025 (25 May Shift) · Mathematics
The sum of three numbers is 20 .
If we multiply the first number by 2 and add the second number to the result, and subtract the third number, we get 23 .
By adding second and third numbers to three times the first number, we get 46.
The above situation can be represented in matrix form as $A X=B$ then the value of determinant formed by the cofactors of matrix A is
- Correct answerA16
- B-4
- C36
- D-64
Solution
Let the three numbers be $x, y, z$. The given conditions are:
$x + y + z = 20$
$2x + y - z = 23$
$3x + y + z = 46$
The matrix $A$ is given by:
$A = \begin{pmatrix} 1 & 1 & 1 \\ 2 & 1 & -1 \\ 3 & 1 & 1 \end{pmatrix}$
The determinant of $A$ is $|A| = 1(1 - (-1)) - 1(2 - (-3)) + 1(2 - 3) = 1(2) - 1(5) + 1(-1) = 2 - 5 - 1 = -4$.
Let $C$ be the matrix of cofactors of $A$. The determinant of the cofactor matrix is given by $|C| = |A|^{n-1}$, where $n$ is the order of the matrix. Here $n = 3$.
$|C| = |A|^{3-1} = |A|^2$.
Substituting $|A| = -4$, we get $|C| = (-4)^2 = 16$.
Answer: 16
$x + y + z = 20$
$2x + y - z = 23$
$3x + y + z = 46$
The matrix $A$ is given by:
$A = \begin{pmatrix} 1 & 1 & 1 \\ 2 & 1 & -1 \\ 3 & 1 & 1 \end{pmatrix}$
The determinant of $A$ is $|A| = 1(1 - (-1)) - 1(2 - (-3)) + 1(2 - 3) = 1(2) - 1(5) + 1(-1) = 2 - 5 - 1 = -4$.
Let $C$ be the matrix of cofactors of $A$. The determinant of the cofactor matrix is given by $|C| = |A|^{n-1}$, where $n$ is the order of the matrix. Here $n = 3$.
$|C| = |A|^{3-1} = |A|^2$.
Substituting $|A| = -4$, we get $|C| = (-4)^2 = 16$.
Answer: 16
Hence, the correct answer is option: 'A'.