# Mathematics Questions and Answers – Determinants – Minors and Cofactors

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This set of Mathematics Multiple Choice Questions & Answers (MCQs) focuses 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

Explanation: 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=$$\begin{vmatrix}1&5&4\\2&3&6\\7&9&4\end{vmatrix}$$?
a) -34
b) 34
c) -17
d) 21

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

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.
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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

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

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.

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

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 Aij is the cofactor of aij.

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

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

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

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

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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