Minors and Cofactors - Class 12 Maths MCQ

This set of Class 12 Maths Chapter 4 Multiple Choice Questions & Answers (MCQs) focuses on “Determinants – Minors and Cofactors”.

1. Which of the following is the formula for cofactor of an element a_ij ?
a) A_ij=(1)^i+j M_ij
b) A_ij=(-2)^i+j M_ij
c) A_ij=(-1)^i+j M_ij
d) A_ij=(-1)^i-j M_ij
View Answer

Answer: c
Explanation: The cofactor of an element a_ij, denoted by A_ij is given by
A_ij=(-1)^i+j M_ij, where M_ij is the minor of the element a_ij.

2. What is the minor of the element 5 in the determinant Δ=\(\begin{vmatrix}1&5&4\\2&3&6\\7&9&4\end{vmatrix}\)?
a) -34
b) 34
c) -17
d) 21
View Answer

Answer: a
Explanation: The minor of element 5 in the determinant Δ=\(\begin{vmatrix}1&5&4\\2&3&6\\7&9&4\end{vmatrix}\) is the determinant obtained by deleting the row and column containing element 5.
∴M₁₂=\(\begin{vmatrix}2&6\\7&4\end{vmatrix}\)=2(4)-7(6)=-34.

3. Find the minor and cofactor respectively for the element 3 in the determinant Δ=\(\begin{vmatrix}1&5\\3&6\end{vmatrix}\).
a) M₂₁=-5, A₂₁=-5
b) M₂₁=5, A₂₁=-5
c) M₂₁=-5, A₂₁=5
d) M₂₁=5, A₂₁=5
View Answer

Answer: b
Explanation: The element 3 is in the second row (i=2) and first column(j=1).
∴M₂₁=5 (obtained by deleting R₂ and C₁ in Δ)
A₂₁=(-1)¹⁺² M₂₁=-1×5 =-5.

4. Find the minor of the element 1 in the determinant Δ=\(\begin{vmatrix}1&5\\3&8\end{vmatrix}\).
a) 5
b) 1
c) 8
d) 3
View Answer

Answer: c
Explanation: The minor of the element 1 can be obtained by deleting the first row and the first column
∴M₁₁=8.

5. Find the cofactor of element -3 in the determinant Δ=\(\begin{vmatrix}1&4&4\\-3&5&9\\2&1&2\end{vmatrix}\).
a) -4
b) 4
c) -5
d) -3
View Answer

Answer: a
Explanation: The minor of element -3 is given by
M₂₁=\(\begin{vmatrix}4&4\\1&2\end{vmatrix}\)=4(2)-4=4 (Obtained by eliminating R₂ and C₁)
∴A₂₁=(-1)²⁺¹ M₂₁=(-1)³ 4=-4.

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6. If Δ=\(\begin{vmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33} \end{vmatrix}\), then the determinant in terms of cofactors A_ij can be expressed as a₁₁ A₁₁+a₂₁ A₂₁+a₃₁ A₃₁.
a) True
b) False
View Answer

Answer: a
Explanation: The given statement is true.
Expanding the determinant Δ=\(\begin{vmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33} \end{vmatrix}\) along R₁, we get
Δ=(-1)¹⁺¹ a₁₁ \(\begin{vmatrix}a_{22}&a_{23}\\a_{32}&a_{33} \end{vmatrix}\)+(-1)¹⁺² a₁₂ \(\begin{vmatrix}a_{21}&a_{23}\\a_{31}&a_{33} \end{vmatrix}\)+(-1)¹⁺³ a₁₃ \(\begin{vmatrix}a_{21}&a_{22}\\a_{31}&a_{32} \end{vmatrix}\)
Δ=a₁₁ A₁₁+a₂₁ A₂₁+a₃₁ A₃₁, where A_ij is the cofactor of a_ij.

7. Find the minor of the element 2 in the determinant Δ=\(\begin{vmatrix}1&9\\2&3\end{vmatrix}\)?
a) 3
b) 9
c) 1
d) 2
View Answer

Answer: b
Explanation: The minor of the element 2 can be obtained by deleting the first row and the first column
∴M₁₁=9.

8. For which of the elements in the determinant Δ=\(\begin{vmatrix}1&8&-6\\2&-3&4\\-7&9&5\end{vmatrix}\) the cofactor is -37.
a) 4
b) 1
c) -6
d) -3
View Answer

Answer: d
Explanation: Consider the element -3 in Δ=\(\begin{vmatrix}1&8&-6\\2&-3&4\\-7&9&5\end{vmatrix}\)
The cofactor of the element -3 is given by
A₂₂=(-1)²⁺² M₂₂
M₂₂=\(\begin{vmatrix}1&-6\\-7&5\end{vmatrix}\)=1(5)-(-6)(-7)=5-42=-37
A₂₂=(-1)²⁺² (-37)=-37.

9. For which of the following elements in the determinant Δ=\(\begin{vmatrix}2&8\\4&7\end{vmatrix}\), the minor of the element is 2?
a) 2
b) 7
c) 4
d) 8
View Answer

Answer: b
Explanation: Consider the element 7 in the determinant Δ=\(\begin{vmatrix}2&8\\4&7\end{vmatrix}\)
The minor of the element 7 can be obtained by deleting R₂ and C₂
∴M₂₂=2
Hence, the minor of the element 7 is 2.

10. For which of the following element in the determinant Δ=\(\begin{vmatrix}5&-5&8\\6&2&-1\\5&-6&8\end{vmatrix}\) , the minor and the cofactor both are zero.
a) -5
b) 2
c) -6
d) 8
View Answer

Answer: b
Explanation: Consider the element 2 in the determinant Δ=\(\begin{vmatrix}5&-5&8\\6&2&-1\\5&-6&8\end{vmatrix}\)
The minor of the element 2 is given by
∴M₂₂=\(\begin{vmatrix}5&8\\5&8\end{vmatrix}\)=40-40=0
⇒A₂₂=(-1)²⁺² (0)=0.

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