RD Sharma Class 12 Solutions - Determinants

 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