Hg
Name: Class 12 STICK TO YOUR WALL IN STUDY AREA
___________________________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________________________
The Hg Classes (8
th
to 12
th
) By: Er Hershit Goyal (B.Tech. IIT BHU), 134-SF, Woodstock Floors, Nirvana Country, Sector 50, GURUGRAM +91 9599697178.
fb.me/thehgclasses linkedin.com/company/the-hg-classes instagram.com/the_hg_classes g.page/the-hg-classes-gurugram thehgclasses.co.in
DETERMINANTS
Definition:
• Determinant of a square matrix A is denoted by |A| or det (A).
• For a square matrix A = [a
11
]
1x1
, |A| = |a
11
| = a
11
• For a square matrix A =
, |A| =
= a
11
x a
22
– a
12
x a
22
• For a square matrix A =
, |A| =
= a1
- b1
+ c1
Properties of Determinants:
For a square matrix A, |A| satisfies the following properties:
• |A’| = |A|
• If any two rows (or columns) of A are interchanged, then sign of |A| changes.
• If any two rows (or columns) are identical or proportional, then |A| = 0.
• If any row (or column) is multiplied by a constant k, then the value of the determinant is also
multiplied by k.
o Multiplying a determinant by k, means multiplying only any one row (or one column) by k.
o If A = [a
ij
]
nxn
, then |k.A| = k
n
|A|
• If elements of a row or a column in a determinant can be expressed as sum of two or more
elements, then the given determinant can be expressed as sum of two or more determinants.
• If to each element of a row or a column of a determinant the equimultiples of corresponding
elements of other rows or columns are added, then value of determinant remains same.
Area of triangle: ∆ = ½
where, (x
1
, y
1
) (x
2
, y
2
) and (x
3
y
3
) are the vertices of a triangle.
Minors and Cofactors:
• Minor of an element a
ij
of a determinant A = M
ij
= determinant obtained by deleting i
th
row and j
th
column.
• Cofactor of an element aij of a determinant A = A
ij
= (-1)
i+j
M
ij
• Thus, by this definition, |A| = a
11
A
11
+ a
12
A
12
+ a
13
A
13
• If elements of one row (or column) are multiplied by the cofactors of any other row (or column),
then their sum is zero. E.g. a
11
A
21
+ a
12
a
22
+ a
13
A
23
Adjoint of a matrix:
• adj A = transpose of matrix of cofactors of A
• (adj A) A = A (adj A) = |A| I, where I is an identity matrix of same order as A.
• A matrix A is called a singular matrix if |A| = 0, and non-singular if |A| ≠ 0
• |AB| = |A||B|
• |adj A| = |A|
n-1
Hg
Name: Class 12 STICK TO YOUR WALL IN STUDY AREA
___________________________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________________________
The Hg Classes (8
th
to 12
th
) By: Er Hershit Goyal (B.Tech. IIT BHU), 134-SF, Woodstock Floors, Nirvana Country, Sector 50, GURUGRAM +91 9599697178.
fb.me/thehgclasses linkedin.com/company/the-hg-classes instagram.com/the_hg_classes g.page/the-hg-classes-gurugram thehgclasses.co.in
Inverse of a matrix:
• If AB = BA = I, then B is called the inverse of A.
• Inverse exists for a non-singular matrix, A
-1
= adj(A) / |A|.
• AA
-1
= A
-1
A = I
• (A
-1
)
-1
= A
• (AB)
-1
= B
-1
A
-1
Solving simultaneous linear equations:
• If a
1
x + b
1
y + c
1
z = d
1
a
2
x + b
2
y + c
2
z = d
2
a
3
x + b
3
y + c
3
z = d
3
,
then, these equations can be written in matrix form as A X = B, where
A =
, X =
, and B =
• The matrix X, the solution matrix, can be obtained by X = A
-1
B, where |A|≠ 0.
• A system of equation is consistent or inconsistent according as its solution
exists or not.
• For a square matrix A in matrix equation AX = B
(i) |A|≠ 0, there exists unique solution, system will be consistent.
(ii) |A| = 0 and (adj A) B ≠ O, (O being a zero matrix), then there exists no solution, system will be
inconsistent.
(iii) |A| = 0 and (adj A) B = O, then system may have infinitely many solutions or no solution, hence,
the system may be consistent or inconsistent respectively.
