Mathematics Multiple Choice Questions & Answers (MCQs) on “Determinants – Minors and Cofactors”.
1. Which of the following is the formula for cofactor of an element aij ?
a) Aij=(1)i+j Mij
b) Aij=(-2)i+j Mij
c) Aij=(-1)i+j Mij
d) Aij=(-1)i-j Mij
Answer: c
Clarification: The cofactor of an element aij, denoted by Aij is given by
Aij=(-1)i+j Mij, where Mij is the minor of the element aij.
2. What is the minor of the element 5 in the determinant Δ=(begin{vmatrix}1&5&4\2&3&6\7&9&4end{vmatrix})?
a) -34
b) 34
c) -17
d) 21
Answer: a
Clarification: The minor of element 5 in the determinant Δ=(begin{vmatrix}1&5&4\2&3&6\7&9&4end{vmatrix}) is the determinant obtained by deleting the row and column containing element 5.
∴M12=(begin{vmatrix}2&6\7&4end{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&6end{vmatrix}).
a) M21=-5, A21=-5
b) M21=5, A21=-5
c) M21=-5, A21=5
d) M21=5, A21=5
Answer: b
Clarification: 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&8end{vmatrix}).
a) 5
b) 1
c) 8
d) 3
Answer: c
Clarification: 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&2end{vmatrix}).
a) -4
b) 4
c) -5
d) -3
Answer: a
Clarification: The minor of element -3 is given by
M21=(begin{vmatrix}4&4\1&2end{vmatrix})=4(2)-4=4 (Obtained by eliminating R2 and C1)
∴A21=(-1)2+1 M21=(-1)3 4=-4.
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 Aij can be expressed as a11 A11+a21 A21+a31 A31.
a) True
b) False
Answer: a
Clarification: 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 Aij is the cofactor of aij.
7. Find the minor of the element 2 in the determinant Δ=(begin{vmatrix}1&9\2&3end{vmatrix})?
a) 3
b) 9
c) 1
d) 2
Answer: b
Clarification: 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&5end{vmatrix}) the cofactor is -37.
a) 4
b) 1
c) -6
d) -3
Answer: d
Clarification: Consider the element -3 in Δ=(begin{vmatrix}1&8&-6\2&-3&4\-7&9&5end{vmatrix})
The cofactor of the element -3 is given by
A22=(-1)2+2 M22
M22=(begin{vmatrix}1&-6\-7&5end{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&7end{vmatrix}), the minor of the element is 2?
a) 2
b) 7
c) 4
d) 8
Answer: b
Clarification: Consider the element 7 in the determinant Δ=(begin{vmatrix}2&8\4&7end{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&8end{vmatrix}) , the minor and the cofactor both are zero.
a) -5
b) 2
c) -6
d) 8