23
Determinants
The determinant is a value associated with a square matrix. It can be IN THIS CHAPTER ....
computed from the entries of the matrix by a specific arithmetic expression, Properties of a Determinant
while other ways to determine its value exist as well. The determinant
Applications of Determinant in
provides important information when the matrix is that of the coefficients of a
Geometry
system of linear equations or when it corresponds to a linear transformation.
Determinants arise in connection with simultaneous linear algebraic Minors and Cofactors of a
equations expressions, which are very complicated and lengthy can be easily Determinant
handled, if they are expressed as determinants. Adjoint of a Square Matrix
An expression which is related to a square matrix Inverse of a Matrix
é a11 a12 K a1n ù Solution of System of Linear
êa a22 K a2n úú Equations by Determinants
A= ê
21
ê M M M ú
ê ú
ë an1 an 2 K ann û
is known as a determinant of A. It is denoted by det( A), D or| A| and written as
a11 a12 K a1n
a21 a22 K a2n
det( A) =| A| =
M M K M
an 1 an 2 K ann
OR
Every square matrix can be associated to an expression or a number which is
known as its determinant.
Value of a Determinant of Order 1
If A = [ a11 ] is a square matrix of order 1 ´ 1,
then | A| = a11
Value of a Determinant of Order 2
éa a12 ù
If A = ê 11 ú is a square matrix of order 2 ´ 2 , then corresponding
ë a21
a22 û
a a12
determinant is| A| = 11 = a11a22 - a12a21
a21 a22
,Determinants 555
Value of a Determinant of Order 3 Properties of a Determinant
é a11 a12 a13 ù
Properties of a determinant of order three only are stated
If A = êê a21 a22 a23 úú is a square matrix of order 3 ´ 3 , below. However, these properties hold for determinants
êë a31 a32 a33 úû of any order. These properties help a good deal in the
evaluation of determinants.
whose corresponding determinant is
(i) The value of the determinant remains unchanged, if
a11 a12 a13 rows are changed into columns and columns are
| A| = a21 a22 a23 changed into rows,
a31 a32 a33 a11 a12 a13 a11 a21 a31
a22 a23 a21 a23 a21 a22 i.e. a21 a22 a23 = a12 a22 a32
\ | A| = a11 - a12 + a13 a31 a32 a33 a13 a23 a33
a32 a33 a31 a33 a31 a32
1 2 3 1 3 2
= a11( a22a33 - a23 a32 ) - a12 ( a21a33 - a23 a31 )
e.g. 3 2 1 = 2 2 3
+ a13 ( a21a32 - a22a31 )
2 3 1 3 1 1
é1 2 3ù 1 2 3
e.g. If A = êê 3 ú
2 1 ú , then| A| = 3 2 1 (ii) If two adjacent rows (or columns) of a determinant
are interchanged, then the value of the
êë 1 1 1 úû 1 1 1 determinant, so obtained is the negative of the value
2 1 3 1 3 2 of the original determinant,
=1 -2 +3 a11 a12 a13 a21 a22 a23
1 1 1 1 1 1
i.e. a21 a22 a23 = - a11 a12 a13
= ( 2 - 1) - 2 ( 3 - 1) + 3 ( 3 - 2)
a31 a32 a33 a31 a32 a33
= 1 - 2 ( 2) + 3
a b c d e f
= 4- 4= 0
e.g. d e f =- a b c
Note The determinant of order greater than 2 can be expanded
g h i g h i
along any row or column.
(iii) If any two rows (or columns) of a determinant are
x sin q cos q identical, then its value is zero,
Example 1. If D1 = - sin q -x 1 a11 a11 a13 a a c
cos q 1 x i.e. a21 a21 a23 = 0 e.g. b b a = 0
x sin 2q cos 2q a31 a31 a33 c c b
and D 2 = - sin 2q -x 1 , x ¹ 0, then for all Here, first and second columns are same.
cos 2q 1 x (iv) Each element of a row (or column) of a determinant
æ pö is multiplied by a constant k, then the value of the
q Î ç0, ÷ new determinant is k times the value of the original
è 2ø (JEE Main 2019)
determinant,
3
(a) D1 + D 2 = - 2( x + x - 1) ka11 ka12 ka13 a11 a12 a13
(b) D1 - D 2 = - 2x3 i.e. a21 a22 a23 = k a21 a22 a23
(c) D1 + D 2 = - 2x3 a31 a32 a33 a31 a32 a33
(d) D1 - D 2 = x(cos 2q - cos 4q ) 2 4 6 1 2 3
x sin q cos q e.g. 4 2 3 =2 4 2 3
Sol. (c) Given determinants are D1 = - sin q - x 1 4 5 6 4 5 6
cos q 1 x
(v) If any two rows (or columns) of a determinants are
= - x3 + sin q cos q - sin q cos q + x cos2 q - x + x sin 2 q proportional, then its value is zero,
= - x3 a11 a12 a13 a11 a12 a13
x sin 2q cos 2q i.e. ka11 ka12 ka13 = k a11 a12 a13 = 0
and D 2 = - sin 2q -x 1 ,x¹0 a31 a32 a33 a31 a32 a33
cos 2q 1 x 2 4 6 1 2 3
= - x3 (similarly as D1) e.g. 1 2 3 =2 1 2 3 =0
So, according to options, we get D1 + D 2 = - 2x3 4 5 6 4 5 6
, 556 JEE Main Mathematics
(vi) If each element of a row (or column) of a determinant f1( x ) f2( x ) f3 ( x )
is the sum of two or more terms, then the (xii) If D( x ) = g1( x ) g2( x ) g3 ( x ) , then
determinant can be expressed as the sum of the two
a b c
or more determinants,
n n n
a11 a12 a13
i.e. a21 + c1 a22 + c2 a23 + c3
å f1( x ) å f2( x ) å f3 ( x )
x =1 x =1 x =1
n n n n
a31 a32 a33
(a) å D( x ) = å g1( x ) å g2( x ) å g3 ( x )
a11 a12 a13 a11 a12 a13 x =1 x =1 x =1 x =1
= a21 a22 a23 + c1 c2 c3 a b c
a31 a32 a33 a31 a32 a33
n n n
a b c a b c a b c
e.g. d + e f+g h+i = d f h + e g i
Õ f1( x ) Õ f2( x ) Õ f3 ( x )
x =1 x =1 x =1
n n n n
j k l j k l j k l
(b) Õ D( x ) = Õ g1( x ) Õ g2( x ) Õ g3 ( x )
(vii) If each element of a row (or column) of a x =1 x =1 x =1 x =1
determinant is multiplied by a constant k and then a b c
added to (or subtracted) from the corresponding
elements of any other row (or column), then the
value of the determinant remains the same,
Important Points
a11 a12 a13
● In D =|aij| is a determinant of order n, then the value
i.e. a21 a22 a23
of the determinant| Aij|, where Aij is the cofactor of aij ,
a31 a32 a33 is Dn - 1.
a11 a12 a13 ● If A = B + C, then it is not necessary that
= a21 a22 a23 det( A) = det( B) + det(C )
a31 + ka21 a32 + ka22 a33 + ka23 ● If A is a square matrix of order n ´ n, then
a b c a b c det( kA) = kn (det A)
e.g. d e f = d e f
● If A, B and C are three square matrix such that ith row
of A is equal to the sum of ith row of B and C and
g h i g + ak h + bk i + ck
remaining rows of A, B and C are same, then
(viii) If each element of a row (or column) of a
det( A) = det( B) + det(C )
determinant is zero, then its value is zero.
● det( An ) = (det A)n , where n is a positive integer.
(ix) If r rows (or r columns) become identical when a is
substituted for x, then ( x - a )r - 1 is a factor of given ● det ( A-1 ) =
1
determinant. det ( A)
(x) The sum of the products of elements of any row ● If the rows and columns are interchange, D is
(or column) of a determinant with the cofactors of unchanged i.e.| AT| =| A|.
the corresponding elements of any other row (or
column) is zero, ● Product of two determinants
a11 a12 a13 i.e. | AB| =| A||B| =|BA| =| ABT| =| AT B|=| AT BT|
i.e. if D = a21 a22 a23 , then
a31 a32 a33 ½x - 4 2 x 2x ½
½
Example 2. If 2 x x - 4 2 x ½ = ( A + Bx)( x - A) 2, then
a11C31 + a12C32 + a13C33 = 0 and so on. ½ ½
½ 2x 2x x - 4½
(xi) The sum of the products of elements of any row
(or column) of a determinant (where | A|n ´ n = D ) the ordered pair ( A, B) is equal to (JEE Main 2018)
with the cofactors of the corresponding elements of (a) ( -4, - 5) (b) ( -4, 3)
same row (or column) is equal to D. (c) ( -4, 5) (d) ( 4, 5)
a11 a12 a13 ½x - 4 2x 2x ½
i.e. D = a21 a22 a23 , then Sol. (c) Given, ½ 2x x - 4 2x ½ = ( A + Bx )( x - A) 2
½ ½
a31 a32 a33 ½ 2x 2x x - 4½
Þ Apply C1 ® C1 + C 2 + C3
a11C11 + a12C12 + a13C13 = D
Determinants
The determinant is a value associated with a square matrix. It can be IN THIS CHAPTER ....
computed from the entries of the matrix by a specific arithmetic expression, Properties of a Determinant
while other ways to determine its value exist as well. The determinant
Applications of Determinant in
provides important information when the matrix is that of the coefficients of a
Geometry
system of linear equations or when it corresponds to a linear transformation.
Determinants arise in connection with simultaneous linear algebraic Minors and Cofactors of a
equations expressions, which are very complicated and lengthy can be easily Determinant
handled, if they are expressed as determinants. Adjoint of a Square Matrix
An expression which is related to a square matrix Inverse of a Matrix
é a11 a12 K a1n ù Solution of System of Linear
êa a22 K a2n úú Equations by Determinants
A= ê
21
ê M M M ú
ê ú
ë an1 an 2 K ann û
is known as a determinant of A. It is denoted by det( A), D or| A| and written as
a11 a12 K a1n
a21 a22 K a2n
det( A) =| A| =
M M K M
an 1 an 2 K ann
OR
Every square matrix can be associated to an expression or a number which is
known as its determinant.
Value of a Determinant of Order 1
If A = [ a11 ] is a square matrix of order 1 ´ 1,
then | A| = a11
Value of a Determinant of Order 2
éa a12 ù
If A = ê 11 ú is a square matrix of order 2 ´ 2 , then corresponding
ë a21
a22 û
a a12
determinant is| A| = 11 = a11a22 - a12a21
a21 a22
,Determinants 555
Value of a Determinant of Order 3 Properties of a Determinant
é a11 a12 a13 ù
Properties of a determinant of order three only are stated
If A = êê a21 a22 a23 úú is a square matrix of order 3 ´ 3 , below. However, these properties hold for determinants
êë a31 a32 a33 úû of any order. These properties help a good deal in the
evaluation of determinants.
whose corresponding determinant is
(i) The value of the determinant remains unchanged, if
a11 a12 a13 rows are changed into columns and columns are
| A| = a21 a22 a23 changed into rows,
a31 a32 a33 a11 a12 a13 a11 a21 a31
a22 a23 a21 a23 a21 a22 i.e. a21 a22 a23 = a12 a22 a32
\ | A| = a11 - a12 + a13 a31 a32 a33 a13 a23 a33
a32 a33 a31 a33 a31 a32
1 2 3 1 3 2
= a11( a22a33 - a23 a32 ) - a12 ( a21a33 - a23 a31 )
e.g. 3 2 1 = 2 2 3
+ a13 ( a21a32 - a22a31 )
2 3 1 3 1 1
é1 2 3ù 1 2 3
e.g. If A = êê 3 ú
2 1 ú , then| A| = 3 2 1 (ii) If two adjacent rows (or columns) of a determinant
are interchanged, then the value of the
êë 1 1 1 úû 1 1 1 determinant, so obtained is the negative of the value
2 1 3 1 3 2 of the original determinant,
=1 -2 +3 a11 a12 a13 a21 a22 a23
1 1 1 1 1 1
i.e. a21 a22 a23 = - a11 a12 a13
= ( 2 - 1) - 2 ( 3 - 1) + 3 ( 3 - 2)
a31 a32 a33 a31 a32 a33
= 1 - 2 ( 2) + 3
a b c d e f
= 4- 4= 0
e.g. d e f =- a b c
Note The determinant of order greater than 2 can be expanded
g h i g h i
along any row or column.
(iii) If any two rows (or columns) of a determinant are
x sin q cos q identical, then its value is zero,
Example 1. If D1 = - sin q -x 1 a11 a11 a13 a a c
cos q 1 x i.e. a21 a21 a23 = 0 e.g. b b a = 0
x sin 2q cos 2q a31 a31 a33 c c b
and D 2 = - sin 2q -x 1 , x ¹ 0, then for all Here, first and second columns are same.
cos 2q 1 x (iv) Each element of a row (or column) of a determinant
æ pö is multiplied by a constant k, then the value of the
q Î ç0, ÷ new determinant is k times the value of the original
è 2ø (JEE Main 2019)
determinant,
3
(a) D1 + D 2 = - 2( x + x - 1) ka11 ka12 ka13 a11 a12 a13
(b) D1 - D 2 = - 2x3 i.e. a21 a22 a23 = k a21 a22 a23
(c) D1 + D 2 = - 2x3 a31 a32 a33 a31 a32 a33
(d) D1 - D 2 = x(cos 2q - cos 4q ) 2 4 6 1 2 3
x sin q cos q e.g. 4 2 3 =2 4 2 3
Sol. (c) Given determinants are D1 = - sin q - x 1 4 5 6 4 5 6
cos q 1 x
(v) If any two rows (or columns) of a determinants are
= - x3 + sin q cos q - sin q cos q + x cos2 q - x + x sin 2 q proportional, then its value is zero,
= - x3 a11 a12 a13 a11 a12 a13
x sin 2q cos 2q i.e. ka11 ka12 ka13 = k a11 a12 a13 = 0
and D 2 = - sin 2q -x 1 ,x¹0 a31 a32 a33 a31 a32 a33
cos 2q 1 x 2 4 6 1 2 3
= - x3 (similarly as D1) e.g. 1 2 3 =2 1 2 3 =0
So, according to options, we get D1 + D 2 = - 2x3 4 5 6 4 5 6
, 556 JEE Main Mathematics
(vi) If each element of a row (or column) of a determinant f1( x ) f2( x ) f3 ( x )
is the sum of two or more terms, then the (xii) If D( x ) = g1( x ) g2( x ) g3 ( x ) , then
determinant can be expressed as the sum of the two
a b c
or more determinants,
n n n
a11 a12 a13
i.e. a21 + c1 a22 + c2 a23 + c3
å f1( x ) å f2( x ) å f3 ( x )
x =1 x =1 x =1
n n n n
a31 a32 a33
(a) å D( x ) = å g1( x ) å g2( x ) å g3 ( x )
a11 a12 a13 a11 a12 a13 x =1 x =1 x =1 x =1
= a21 a22 a23 + c1 c2 c3 a b c
a31 a32 a33 a31 a32 a33
n n n
a b c a b c a b c
e.g. d + e f+g h+i = d f h + e g i
Õ f1( x ) Õ f2( x ) Õ f3 ( x )
x =1 x =1 x =1
n n n n
j k l j k l j k l
(b) Õ D( x ) = Õ g1( x ) Õ g2( x ) Õ g3 ( x )
(vii) If each element of a row (or column) of a x =1 x =1 x =1 x =1
determinant is multiplied by a constant k and then a b c
added to (or subtracted) from the corresponding
elements of any other row (or column), then the
value of the determinant remains the same,
Important Points
a11 a12 a13
● In D =|aij| is a determinant of order n, then the value
i.e. a21 a22 a23
of the determinant| Aij|, where Aij is the cofactor of aij ,
a31 a32 a33 is Dn - 1.
a11 a12 a13 ● If A = B + C, then it is not necessary that
= a21 a22 a23 det( A) = det( B) + det(C )
a31 + ka21 a32 + ka22 a33 + ka23 ● If A is a square matrix of order n ´ n, then
a b c a b c det( kA) = kn (det A)
e.g. d e f = d e f
● If A, B and C are three square matrix such that ith row
of A is equal to the sum of ith row of B and C and
g h i g + ak h + bk i + ck
remaining rows of A, B and C are same, then
(viii) If each element of a row (or column) of a
det( A) = det( B) + det(C )
determinant is zero, then its value is zero.
● det( An ) = (det A)n , where n is a positive integer.
(ix) If r rows (or r columns) become identical when a is
substituted for x, then ( x - a )r - 1 is a factor of given ● det ( A-1 ) =
1
determinant. det ( A)
(x) The sum of the products of elements of any row ● If the rows and columns are interchange, D is
(or column) of a determinant with the cofactors of unchanged i.e.| AT| =| A|.
the corresponding elements of any other row (or
column) is zero, ● Product of two determinants
a11 a12 a13 i.e. | AB| =| A||B| =|BA| =| ABT| =| AT B|=| AT BT|
i.e. if D = a21 a22 a23 , then
a31 a32 a33 ½x - 4 2 x 2x ½
½
Example 2. If 2 x x - 4 2 x ½ = ( A + Bx)( x - A) 2, then
a11C31 + a12C32 + a13C33 = 0 and so on. ½ ½
½ 2x 2x x - 4½
(xi) The sum of the products of elements of any row
(or column) of a determinant (where | A|n ´ n = D ) the ordered pair ( A, B) is equal to (JEE Main 2018)
with the cofactors of the corresponding elements of (a) ( -4, - 5) (b) ( -4, 3)
same row (or column) is equal to D. (c) ( -4, 5) (d) ( 4, 5)
a11 a12 a13 ½x - 4 2x 2x ½
i.e. D = a21 a22 a23 , then Sol. (c) Given, ½ 2x x - 4 2x ½ = ( A + Bx )( x - A) 2
½ ½
a31 a32 a33 ½ 2x 2x x - 4½
Þ Apply C1 ® C1 + C 2 + C3
a11C11 + a12C12 + a13C13 = D
