Question:
The co-factors of the elements of second column of $\left[\begin{array}{ccc}1 & -1 & 2 \\ 3 & 2 & 1 \\ -1 & 3 & 4\end{array}\right]$ are
The co-factors of the elements of second column of $\left[\begin{array}{ccc}1 & -1 & 2 \\ 3 & 2 & 1 \\ -1 & 3 & 4\end{array}\right]$ are
Show Hint
Remember the alternating sign pattern for a $3 \times 3$ matrix: the second column signs are always $(-, +, -)$. Calculate the minors and simply apply these signs to get the cofactors quickly.
Updated On: Jun 4, 2026
- -13, 6, 5
- 13, 5, 6
- 13, -6, -5
- -13, -6, 5
Show Solution
Verified By Collegedunia
The Correct Option is A
Solution and Explanation
Step 1: Understanding the Question:
We need to calculate the cofactors specifically for the elements located in the second column of the given $3 \times 3$ matrix.
Step 2: Key Formula or Approach:
The cofactor $C_{ij}$ of an element $a_{ij}$ in a matrix is given by:
$$C_{ij} = (-1)^{i+j} M_{ij}$$ where $M_{ij}$ is the minor of the element (the determinant of the $2 \times 2$ matrix left after removing the $i$-th row and $j$-th column).
Step 3: Detailed Explanation:
Let the given matrix be $A$. The elements of the second column are $a_{12} = -1$, $a_{22} = 2$, and $a_{32} = 3$.
6. Cofactor of $a_{12}$:
$$C_{12} = (-1)^{1+2} \begin{vmatrix} 3 & 1 \\ -1 & 4 \end{vmatrix} = -1 [(3)(4) - (1)(-1)] = -[12 + 1] = -13$$ 7. Cofactor of $a_{22}$:
$$C_{22} = (-1)^{2+2} \begin{vmatrix} 1 & 2 \\ -1 & 4 \end{vmatrix} = +1 [(1)(4) - (2)(-1)] = +[4 + 2] = 6$$ 8. Cofactor of $a_{32}$:
$$C_{32} = (-1)^{3+2} \begin{vmatrix} 1 & 2 \\ 3 & 1 \end{vmatrix} = -1 [(1)(1) - (2)(3)] = -[1 - 6] = -[-5] = 5$$ The cofactors are $-13, 6$, and $5$.
Step 4: Final Answer:
The cofactors are $-13, 6, 5$, matching option (A).
We need to calculate the cofactors specifically for the elements located in the second column of the given $3 \times 3$ matrix.
Step 2: Key Formula or Approach:
The cofactor $C_{ij}$ of an element $a_{ij}$ in a matrix is given by:
$$C_{ij} = (-1)^{i+j} M_{ij}$$ where $M_{ij}$ is the minor of the element (the determinant of the $2 \times 2$ matrix left after removing the $i$-th row and $j$-th column).
Step 3: Detailed Explanation:
Let the given matrix be $A$. The elements of the second column are $a_{12} = -1$, $a_{22} = 2$, and $a_{32} = 3$.
6. Cofactor of $a_{12}$:
$$C_{12} = (-1)^{1+2} \begin{vmatrix} 3 & 1 \\ -1 & 4 \end{vmatrix} = -1 [(3)(4) - (1)(-1)] = -[12 + 1] = -13$$ 7. Cofactor of $a_{22}$:
$$C_{22} = (-1)^{2+2} \begin{vmatrix} 1 & 2 \\ -1 & 4 \end{vmatrix} = +1 [(1)(4) - (2)(-1)] = +[4 + 2] = 6$$ 8. Cofactor of $a_{32}$:
$$C_{32} = (-1)^{3+2} \begin{vmatrix} 1 & 2 \\ 3 & 1 \end{vmatrix} = -1 [(1)(1) - (2)(3)] = -[1 - 6] = -[-5] = 5$$ The cofactors are $-13, 6$, and $5$.
Step 4: Final Answer:
The cofactors are $-13, 6, 5$, matching option (A).
Was this answer helpful?
0
0
Top MHT CET Mathematics Questions
- The line $5x + y - 1 = 0$ coincides with one of the lines given by $5x^2 + xy - kx - 2y + 2 = 0 $ then the value of k is
- If $\int\frac{f\left(x\right)}{log \left(sin\,x\right)}dx = log\left[log\,sin\,x\right]+c$ then $f\left(x\right)=$
- If $\int\limits^{K}_0 \frac{dx}{2 + 18 x^2} = \frac{\pi}{24}$, then the value of K is
- If $\int^{\pi/2}_{0} \log\cos x dx =\frac{\pi}{2} \log\left(\frac{1}{2}\right)$ then $ \int^{\pi/2}_{0} \log\sec x dx = $
- The point on the curve $y = \sqrt{x - 1}$ where the tangent is perpendicular to the line $2x + y - 5 = 0 $ is
View More Questions
Top MHT CET Determinants Questions
- If Matrix $A = \begin{bmatrix}1&2\\ 4&3\end{bmatrix}$ such that $Ax = I$, then $X = $_______
- If $A=\begin{bmatrix}1&1&0\\ 2&1&5\\ 1&2&1\end{bmatrix}$, then $a_{11}A_{21} +a_{12}A_{22}+a_{13}A_{23} $ is equal to
- Matrix $A = \begin{bmatrix}1&2&3\\ 1&1&5\\ 2&4&7\end{bmatrix}$then the value of $a_{31} A_{31} + a_{32} A_{32} + a_{33 } + A_{33} $ is
- If \( f(x) \) is differentiable at \( x = 1 \) and \[ \lim_{h \to 0} \frac{1}{h} f(1 + h) = 5, \] then \( f'(1) \) is equal to:
- If \( A = \begin{bmatrix} 2 & 1
3 & 4 \end{bmatrix} \), then the determinant of matrix \( A \) is:
View More Questions
Top MHT CET Questions
- During r- DNA technology, which one of the following enzymes is used for cleaving DNA molecule ?
- In non uniform circular motion, the ratio of tangential to radial acceleration is (r = radius of circle, $v =$ speed of the particle, $\alpha =$ angular acceleration)
- The temperature of $32^{\circ}C$ is equivalent to
- The line $5x + y - 1 = 0$ coincides with one of the lines given by $5x^2 + xy - kx - 2y + 2 = 0 $ then the value of k is
- The heat of formation of water is $ 260\, kJ $ . How much $ H_2O $ is decomposed by $ 130\, kJ $ of heat ?
View More Questions