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 Page 1


6. Determinants
Exercise 6.1
1 A. Question
Write the minors and cofactors of each element of the first column of the following matrices and hence
evaluate the determinant in each case:
Answer
Let M
ij
 and C
ij
 represents the minor and co–factor of an element, where i and j represent the row and
column.
The minor of the matrix can be obtained for a particular element by removing the row and column where the
element is present. Then finding the absolute value of the matrix newly formed.
Also, C
ij
 = (–1)
i+j
 × M
ij
M
11
 = –1
M
21
 = 20
C
11
 = (–1)
1+1
 × M
11
= 1 × –1
= –1
C
21
 = (–1)
2+1
 × M
21
= 20 × –1
= –20
Now expanding along the first column we get
|A| = a
11
 × C
11
 + a
21
× C
21
= 5× (–1) + 0 × (–20)
= –5
1 B. Question
Write the minors and cofactors of each element of the first column of the following matrices and hence
evaluate the determinant in each case:
Answer
Let M
ij
 and C
ij
 represents the minor and co–factor of an element, where i and j represent the row and
column.
The minor of matrix can be obtained for particular element by removing the row and column where the
element is present. Then finding the absolute value of the matrix newly formed.
Also, C
ij
 = (–1)
i+j
 × M
ij
Page 2


6. Determinants
Exercise 6.1
1 A. Question
Write the minors and cofactors of each element of the first column of the following matrices and hence
evaluate the determinant in each case:
Answer
Let M
ij
 and C
ij
 represents the minor and co–factor of an element, where i and j represent the row and
column.
The minor of the matrix can be obtained for a particular element by removing the row and column where the
element is present. Then finding the absolute value of the matrix newly formed.
Also, C
ij
 = (–1)
i+j
 × M
ij
M
11
 = –1
M
21
 = 20
C
11
 = (–1)
1+1
 × M
11
= 1 × –1
= –1
C
21
 = (–1)
2+1
 × M
21
= 20 × –1
= –20
Now expanding along the first column we get
|A| = a
11
 × C
11
 + a
21
× C
21
= 5× (–1) + 0 × (–20)
= –5
1 B. Question
Write the minors and cofactors of each element of the first column of the following matrices and hence
evaluate the determinant in each case:
Answer
Let M
ij
 and C
ij
 represents the minor and co–factor of an element, where i and j represent the row and
column.
The minor of matrix can be obtained for particular element by removing the row and column where the
element is present. Then finding the absolute value of the matrix newly formed.
Also, C
ij
 = (–1)
i+j
 × M
ij
M
11
 = 3
M
21
 = 4
C
11
 = (–1)
1+1
 × M
11
= 1 × 3
= 3
C
21
 = (–1)
2+1
 × 4
= –1 × 4
= –4
Now expanding along the first column we get
|A| = a
11
 × C
11
 + a
21
× C
21
= –1× 3 + 2 × (–4)
= –11
1 C. Question
Write the minors and cofactors of each element of the first column of the following matrices and hence
evaluate the determinant in each case:
Answer
Let M
ij
 and C
ij
 represents the minor and co–factor of an element, where i and j represent the row and
column.
The minor of the matrix can be obtained for a particular element by removing the row and column where the
element is present. Then finding the absolute value of the matrix newly formed.
Also, C
ij
 = (–1)
i+j
 × M
ij
M
11
 = –1×2 – 5×2
M
11
 = –12
M
21
 = –3×2 – 5×2
M
21
 = –16
M
31
 = –3×2 – (–1) × 2
Page 3


6. Determinants
Exercise 6.1
1 A. Question
Write the minors and cofactors of each element of the first column of the following matrices and hence
evaluate the determinant in each case:
Answer
Let M
ij
 and C
ij
 represents the minor and co–factor of an element, where i and j represent the row and
column.
The minor of the matrix can be obtained for a particular element by removing the row and column where the
element is present. Then finding the absolute value of the matrix newly formed.
Also, C
ij
 = (–1)
i+j
 × M
ij
M
11
 = –1
M
21
 = 20
C
11
 = (–1)
1+1
 × M
11
= 1 × –1
= –1
C
21
 = (–1)
2+1
 × M
21
= 20 × –1
= –20
Now expanding along the first column we get
|A| = a
11
 × C
11
 + a
21
× C
21
= 5× (–1) + 0 × (–20)
= –5
1 B. Question
Write the minors and cofactors of each element of the first column of the following matrices and hence
evaluate the determinant in each case:
Answer
Let M
ij
 and C
ij
 represents the minor and co–factor of an element, where i and j represent the row and
column.
The minor of matrix can be obtained for particular element by removing the row and column where the
element is present. Then finding the absolute value of the matrix newly formed.
Also, C
ij
 = (–1)
i+j
 × M
ij
M
11
 = 3
M
21
 = 4
C
11
 = (–1)
1+1
 × M
11
= 1 × 3
= 3
C
21
 = (–1)
2+1
 × 4
= –1 × 4
= –4
Now expanding along the first column we get
|A| = a
11
 × C
11
 + a
21
× C
21
= –1× 3 + 2 × (–4)
= –11
1 C. Question
Write the minors and cofactors of each element of the first column of the following matrices and hence
evaluate the determinant in each case:
Answer
Let M
ij
 and C
ij
 represents the minor and co–factor of an element, where i and j represent the row and
column.
The minor of the matrix can be obtained for a particular element by removing the row and column where the
element is present. Then finding the absolute value of the matrix newly formed.
Also, C
ij
 = (–1)
i+j
 × M
ij
M
11
 = –1×2 – 5×2
M
11
 = –12
M
21
 = –3×2 – 5×2
M
21
 = –16
M
31
 = –3×2 – (–1) × 2
M
31
 = –4
C
11
 = (–1)
1+1
 × M
11
= 1 × –12
= –12
C
21
 = (–1)
2+1
 × M
21
= –1 × –16
= 16
C
31
 = (–1)
3+1
 × M
31
= 1 × –4
= –4
Now expanding along the first column we get
|A| = a
11
 × C
11
 + a
21
× C
21
+ a
31
× C
31
= 1× (–12) + 4 × 16 + 3× (–4)
= –12 + 64 –12
= 40
1 D. Question
Write the minors and cofactors of each element of the first column of the following matrices and hence
evaluate the determinant in each case:
Answer
Let M
ij
 and C
ij
 represents the minor and co–factor of an element, where i and j represent the row and
column.
The minor of the matrix can be obtained for a particular element by removing the row and column where the
element is present. Then finding the absolute value of the matrix newly formed.
Also, C
ij
 = (–1)
i+j
 × M
ij
M
11
 = b × ab – c × ca
M
11
 = ab
2
 – ac
2
M
21
 = a × ab – c × bc
M
21
 = a
2
b – c
2
b
Page 4


6. Determinants
Exercise 6.1
1 A. Question
Write the minors and cofactors of each element of the first column of the following matrices and hence
evaluate the determinant in each case:
Answer
Let M
ij
 and C
ij
 represents the minor and co–factor of an element, where i and j represent the row and
column.
The minor of the matrix can be obtained for a particular element by removing the row and column where the
element is present. Then finding the absolute value of the matrix newly formed.
Also, C
ij
 = (–1)
i+j
 × M
ij
M
11
 = –1
M
21
 = 20
C
11
 = (–1)
1+1
 × M
11
= 1 × –1
= –1
C
21
 = (–1)
2+1
 × M
21
= 20 × –1
= –20
Now expanding along the first column we get
|A| = a
11
 × C
11
 + a
21
× C
21
= 5× (–1) + 0 × (–20)
= –5
1 B. Question
Write the minors and cofactors of each element of the first column of the following matrices and hence
evaluate the determinant in each case:
Answer
Let M
ij
 and C
ij
 represents the minor and co–factor of an element, where i and j represent the row and
column.
The minor of matrix can be obtained for particular element by removing the row and column where the
element is present. Then finding the absolute value of the matrix newly formed.
Also, C
ij
 = (–1)
i+j
 × M
ij
M
11
 = 3
M
21
 = 4
C
11
 = (–1)
1+1
 × M
11
= 1 × 3
= 3
C
21
 = (–1)
2+1
 × 4
= –1 × 4
= –4
Now expanding along the first column we get
|A| = a
11
 × C
11
 + a
21
× C
21
= –1× 3 + 2 × (–4)
= –11
1 C. Question
Write the minors and cofactors of each element of the first column of the following matrices and hence
evaluate the determinant in each case:
Answer
Let M
ij
 and C
ij
 represents the minor and co–factor of an element, where i and j represent the row and
column.
The minor of the matrix can be obtained for a particular element by removing the row and column where the
element is present. Then finding the absolute value of the matrix newly formed.
Also, C
ij
 = (–1)
i+j
 × M
ij
M
11
 = –1×2 – 5×2
M
11
 = –12
M
21
 = –3×2 – 5×2
M
21
 = –16
M
31
 = –3×2 – (–1) × 2
M
31
 = –4
C
11
 = (–1)
1+1
 × M
11
= 1 × –12
= –12
C
21
 = (–1)
2+1
 × M
21
= –1 × –16
= 16
C
31
 = (–1)
3+1
 × M
31
= 1 × –4
= –4
Now expanding along the first column we get
|A| = a
11
 × C
11
 + a
21
× C
21
+ a
31
× C
31
= 1× (–12) + 4 × 16 + 3× (–4)
= –12 + 64 –12
= 40
1 D. Question
Write the minors and cofactors of each element of the first column of the following matrices and hence
evaluate the determinant in each case:
Answer
Let M
ij
 and C
ij
 represents the minor and co–factor of an element, where i and j represent the row and
column.
The minor of the matrix can be obtained for a particular element by removing the row and column where the
element is present. Then finding the absolute value of the matrix newly formed.
Also, C
ij
 = (–1)
i+j
 × M
ij
M
11
 = b × ab – c × ca
M
11
 = ab
2
 – ac
2
M
21
 = a × ab – c × bc
M
21
 = a
2
b – c
2
b
M
31
 = a × ca – b × bc
M
31
 = a
2
c – b
2
c
C
11
 = (–1)
1+1
 × M
11
= 1 × (ab
2
 – ac
2
)
= ab
2
 – ac
2
C
21
 = (–1)
2+1
 × M
21
= –1 × (a
2
b – c
2
b)
= c
2
b – a
2
b
C
31
 = (–1)
3+1
 × M
31
= 1 × (a
2
c – b
2
c)
= a
2
c – b
2
c
Now expanding along the first column we get
|A| = a
11
 × C
11
 + a
21
× C
21
+ a
31
× C
31
= 1× (ab
2
 – ac
2
) + 1 × (c
2
b – a
2
b) + 1× (a
2
c – b
2
c)
= ab
2
 – ac
2
 + c
2
b – a
2
b + a
2
c – b
2
c
1 E. Question
Write the minors and cofactors of each element of the first column of the following matrices and hence
evaluate the determinant in each case:
Answer
Let M
ij
 and C
ij
 represents the minor and co–factor of an element, where i and j represent the row and
column.
The minor of matrix can be obtained for particular element by removing the row and column where the
element is present. Then finding the absolute value of the matrix newly formed.
Also, C
ij
 = (–1)
i+j
 × M
ij
M
11
 = 5×1 – 7×0
M
11
 = 5
M
21
 = 2×1 – 7×6
Page 5


6. Determinants
Exercise 6.1
1 A. Question
Write the minors and cofactors of each element of the first column of the following matrices and hence
evaluate the determinant in each case:
Answer
Let M
ij
 and C
ij
 represents the minor and co–factor of an element, where i and j represent the row and
column.
The minor of the matrix can be obtained for a particular element by removing the row and column where the
element is present. Then finding the absolute value of the matrix newly formed.
Also, C
ij
 = (–1)
i+j
 × M
ij
M
11
 = –1
M
21
 = 20
C
11
 = (–1)
1+1
 × M
11
= 1 × –1
= –1
C
21
 = (–1)
2+1
 × M
21
= 20 × –1
= –20
Now expanding along the first column we get
|A| = a
11
 × C
11
 + a
21
× C
21
= 5× (–1) + 0 × (–20)
= –5
1 B. Question
Write the minors and cofactors of each element of the first column of the following matrices and hence
evaluate the determinant in each case:
Answer
Let M
ij
 and C
ij
 represents the minor and co–factor of an element, where i and j represent the row and
column.
The minor of matrix can be obtained for particular element by removing the row and column where the
element is present. Then finding the absolute value of the matrix newly formed.
Also, C
ij
 = (–1)
i+j
 × M
ij
M
11
 = 3
M
21
 = 4
C
11
 = (–1)
1+1
 × M
11
= 1 × 3
= 3
C
21
 = (–1)
2+1
 × 4
= –1 × 4
= –4
Now expanding along the first column we get
|A| = a
11
 × C
11
 + a
21
× C
21
= –1× 3 + 2 × (–4)
= –11
1 C. Question
Write the minors and cofactors of each element of the first column of the following matrices and hence
evaluate the determinant in each case:
Answer
Let M
ij
 and C
ij
 represents the minor and co–factor of an element, where i and j represent the row and
column.
The minor of the matrix can be obtained for a particular element by removing the row and column where the
element is present. Then finding the absolute value of the matrix newly formed.
Also, C
ij
 = (–1)
i+j
 × M
ij
M
11
 = –1×2 – 5×2
M
11
 = –12
M
21
 = –3×2 – 5×2
M
21
 = –16
M
31
 = –3×2 – (–1) × 2
M
31
 = –4
C
11
 = (–1)
1+1
 × M
11
= 1 × –12
= –12
C
21
 = (–1)
2+1
 × M
21
= –1 × –16
= 16
C
31
 = (–1)
3+1
 × M
31
= 1 × –4
= –4
Now expanding along the first column we get
|A| = a
11
 × C
11
 + a
21
× C
21
+ a
31
× C
31
= 1× (–12) + 4 × 16 + 3× (–4)
= –12 + 64 –12
= 40
1 D. Question
Write the minors and cofactors of each element of the first column of the following matrices and hence
evaluate the determinant in each case:
Answer
Let M
ij
 and C
ij
 represents the minor and co–factor of an element, where i and j represent the row and
column.
The minor of the matrix can be obtained for a particular element by removing the row and column where the
element is present. Then finding the absolute value of the matrix newly formed.
Also, C
ij
 = (–1)
i+j
 × M
ij
M
11
 = b × ab – c × ca
M
11
 = ab
2
 – ac
2
M
21
 = a × ab – c × bc
M
21
 = a
2
b – c
2
b
M
31
 = a × ca – b × bc
M
31
 = a
2
c – b
2
c
C
11
 = (–1)
1+1
 × M
11
= 1 × (ab
2
 – ac
2
)
= ab
2
 – ac
2
C
21
 = (–1)
2+1
 × M
21
= –1 × (a
2
b – c
2
b)
= c
2
b – a
2
b
C
31
 = (–1)
3+1
 × M
31
= 1 × (a
2
c – b
2
c)
= a
2
c – b
2
c
Now expanding along the first column we get
|A| = a
11
 × C
11
 + a
21
× C
21
+ a
31
× C
31
= 1× (ab
2
 – ac
2
) + 1 × (c
2
b – a
2
b) + 1× (a
2
c – b
2
c)
= ab
2
 – ac
2
 + c
2
b – a
2
b + a
2
c – b
2
c
1 E. Question
Write the minors and cofactors of each element of the first column of the following matrices and hence
evaluate the determinant in each case:
Answer
Let M
ij
 and C
ij
 represents the minor and co–factor of an element, where i and j represent the row and
column.
The minor of matrix can be obtained for particular element by removing the row and column where the
element is present. Then finding the absolute value of the matrix newly formed.
Also, C
ij
 = (–1)
i+j
 × M
ij
M
11
 = 5×1 – 7×0
M
11
 = 5
M
21
 = 2×1 – 7×6
M
21
 = –40
M
31
 = 2×0 – 5×6
M
31
 = –30
C
11
 = (–1)
1+1
 × M
11
= 1 × 5
= 5
C
21
 = (–1)
2+1
 × M
21
= –1 × –40
= 40
C
31
 = (–1)
3+1
 × M
31
= 1 × –30
= –30
Now expanding along the first column we get
|A| = a
11
 × C
11
 + a
21
× C
21
+ a
31
× C
31
= 0× 5 + 1 × 40 + 3× (–30)
= 0 + 40 – 90
= 50
1 F. Question
Write the minors and cofactors of each element of the first column of the following matrices and hence
evaluate the determinant in each case:
Answer
Let M
ij
 and C
ij
 represents the minor and co–factor of an element, where i and j represent the row and
column.
The minor of matrix can be obtained for particular element by removing the row and column where the
element is present. Then finding the absolute value of the matrix newly formed.
Also, C
ij
 = (–1)
i+j
 × M
ij
M
11
 = b × c – f × f
M
11
 = bc– f
2

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The "RD Sharma Class 12 Solutions - Determinants JEE Questions" guide is a valuable resource for all aspiring students preparing for the JEE exam. It focuses on providing a wide range of practice questions to help students gauge their understanding of the exam topics. These questions cover the entire syllabus, ensuring comprehensive preparation. The guide includes previous years' question papers for students to familiarize themselves with the exam's format and difficulty level. Additionally, it offers subject-specific question banks, allowing students to focus on weak areas and improve their performance.

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