Download Determinant of a nxn matrix

Important notes on Determinants
1. The Determinant of a Matrix

DEFINITION: Determinants play an important role in finding the inverse of a
matrix and also in solving systems of linear equations. The determinant of a
square matrix A is a number associated with every square matrix and is denoted
by det(A) or |A|.
 MINOR The minor of the element aij of |A| is given by Mij, where Mij is the determinant
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of the (n-1) x (n-1) matrix that is obtained by deleting row i and column j (where the
element aij lies) of the determinant of A.
COFACTOR - The cofactor of the element aij given by Aij = (-1)i+j Mij
+ − +
+ −
Sign convention for the cofactors [
] , [− + −] etc.
− +
+ − +
ADJOINT (OR ADJUGATE) OF A MATRIX - the transpose of the matrix of cofactors
adj(A) =[Cij]T=[Cji]
 If A is a square matrix of order n then A . adj A = adj A. A = |𝐴|𝐼𝑛
 If A is a square matrix of order n then|𝑎𝑑𝑗𝐴| = |𝐴|𝑛−1
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Determinant of a 2x2 matrix
If A =
𝒂𝟏𝟏
[𝒂𝒊𝒋]𝟐×𝟐 = [𝒂
𝟐𝟏
𝒂𝟏𝟐
𝒂𝟐𝟐 ] then|𝑨| = 𝒂𝟏𝟏 𝒂𝟐𝟐 − 𝒂𝟐𝟏 𝒂𝟏𝟐
𝒂𝟏𝟏 𝒂𝟏𝟐 𝒂𝟏𝟑
Determinant of a 3x3 matrixIf 𝑨 = [𝒂𝒊𝒋 ]𝟑×𝟑 = [𝒂𝟐𝟏 𝒂𝟐𝟐 𝒂𝟐𝟑 ] then
𝒂𝟑𝟏 𝒂𝟑𝟐 𝒂𝟑𝟑
𝒂𝟐𝟐 𝒂𝟐𝟑
𝒂𝟐𝟏 𝒂𝟐𝟑
𝒂𝟐𝟏 𝒂𝟐𝟐
det A= |𝑨| = 𝒂𝟏𝟏 |𝒂
| − 𝒂𝟏𝟐 |𝒂
| + 𝒂𝟏𝟑 |𝒂
|
𝒂
𝒂
𝟑𝟐
𝟑𝟑
𝟑𝟏
𝟑𝟑
𝟑𝟏 𝒂𝟑𝟐
A= |𝑨| = 𝒂𝟏𝟏 𝑨𝟏𝟏 + 𝒂𝟏𝟐 𝑨𝟏𝟐 +
⋯ + 𝒂𝟏𝒏 𝑨𝟏𝒏 where Aij is the cofactor of the element aij given by Aij = (-1)i+j Mij.
Singular matrix – A square matrix is said to be singular if |𝑨| = 𝟎
Non- Singular matrix – A square matrix is said to be non-singular if |𝑨| ≠ 𝟎
Determinant of a nxn matrix If A =
[𝒂𝒊𝒋]𝒏×𝒏 then det
If A and B are non-singular matrices of the same order then AB and BA are also
non-singular matrices of the same order.
2. THE PROPERTIES OF DETERMINANTS
P1
P2
P3
P4
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P6
P7
The value of the determinant remains unchanged if its rows and columns are interchanged
𝑅𝑖 ⟷ 𝐶𝑖 ⟹ ∆= ∆′ ⇒ |𝐴| = |𝐴𝑇 |
If two rows or columns of a determinant is interchanged the sign of the determinant changes
𝑅𝑖 ⟷ 𝑅𝑗 or 𝐶𝑖 ⟷ 𝐶𝑗 ⟹ ∆= −∆.
If any two rows or columns of a determinant is identical then its value is 0.
If each element of any row or column is multiplied by a scalar then the value of the
determinant gets multiplied by that scalar.𝑅𝑖 ⟶ 𝑘𝑅𝑖 𝑜𝑟 𝐶𝑖 ⟶ 𝑘𝐶𝑖 ⟹ ∆= 𝑘∆.
If each element of a row or column can be expressed as sum of two or more terms then the
determinant can be expressed as sum of two or more determinants of the same order.
If any row (or column) of a determinant is proportional to any other row (or column) then
the value of the determinant is 0. 𝑖. 𝑒. 𝑅𝑖 = 𝑘𝑅𝑗 𝑜𝑟 𝐶𝑖 = 𝑘𝐶𝑗 ⟹ ∆= 0.
If ,to each element of any row or column, is added the equimultiples of the corresponding
elements of one or more rows or columns, the value of the determinant remains unchanged.
𝑅𝑖 ⟶ 𝑅𝑖 + 𝑘𝑅𝑗 𝑜𝑟 𝐶𝑖 ⟶ 𝐶𝑖 +𝑘𝐶𝑗 ⟹ ∆1 = ∆.
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If a determinant can be regarded as a polynomial function in x , and if it becomes 0 by
putting x = a then (x – a) is a factor of the determinant.
If the elements of any row or column is multiplied by its corresponding cofactors and
summed up then the result is the determinant itself.
If the elements of any row or column is multiplied by the cofactors of any other row or
column and summed up then the result is 0.
The determinant of the product of two square matrices of the same order is equal to the
product of their determinants. i.e. |𝐴𝐵| = |𝐴||𝐵|
If each element of a particular row or column is 0 then the value of the determinant is 0.
If A is a square matrix of order n then |𝑘𝐴| = 𝑘 𝑛 |𝐴|
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3. APPLICATION OF DETERMINANT
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𝑥1 𝑦1 1
Area of the triangle whose vertices are (x1,y1) , (x2 , y2), (x3, y3) is given by ∆= | |𝑥2 𝑦2 1||
2
𝑥3 𝑦3 1
𝑥1 𝑦1 1
𝑥
Condition of collinearity of the points (x1,y1) , (x2 , y2), (x3, y3) is given by | 2 𝑦2 1| = 0
𝑥3 𝑦3 1
Equation of a line passing through the points(x, y),(x1,y1) ,
(x2 , y2) is given by
𝑥 𝑦 1
|𝑥1 𝑦1 1| = 0
𝑥2 𝑦2 1
1
4. The Inverse of a Matrix
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DEFINITION: If A = [𝒂𝒊𝒋]𝒏×𝒏is non-singular ( i.e. det(A) does not equal zero ),
then there exists an nxn matrix A-1 which is called the inverse of A, such that:
𝐴−1 =
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𝐴𝑑𝑗 𝐴
|𝐴|
AA-1= A-1A = I where I is the identity matrix.
If A and B are two invertible matrices of the same order ,then (AB) -1= B-1A-1
If A , B and C are three invertible matrices of the same order ,then (ABC) -1= C-1B-1A-1
If A is an invertible matrix then AT is also invertible and (AT) -1 = (A-1) T
5. SOLVING SYSTEMS OF EQUATIONS USING INVERSE MATRIX METHOD
DEFINITION: A system of linear equations is a set of equations with n equations and
n unknowns, is of the form of
The unknowns are denoted by x1,x2,...xn and the
coefficients (a's and b's above) are assumed to be given. In matrix form the system of
equations above can be written as:
which can be expressed in matrix equation as
AX =B
By pre-multiplying both sides of this equation by A-1 gives:
A-1 (AX) = A-1 B ⟹ (𝐴−1 𝐴)𝑋 = 𝐴−1 𝐵 ⇒ 𝐼𝑋 = 𝐴−1 𝐵 ⇒ 𝑋 = 𝐴−1 𝐵
STEPS
i. Evaluate |𝐴|. Refer the note given below.
ii. Evaluate the cofactors of elements of A.
iii. Form the adjoint of A as the matrix of cofactors
𝐴𝑑𝑗 𝐴
iv. Calculate 𝐴−1 = |𝐴|
NOTE - From the above it is clear that the existence of a solution depends on the value
of the determinant of A. There are three cases:
1. If the |𝐴| ≠ 0 then the system is consistent with unique solution given by 𝑋 =
𝐴−1 𝐵
2. If |𝐴| = 0 (A is singular) and adjA .B ≠ 0 then the solution does not exist. The
system is inconsistent.
3. If |𝐴| = 0 (A is singular) and adjA .B = 0 then the system is consistent with
infinitely many solutions.to find these solutions put z = k in two of the equations and
solve them by matrix method.
6. For homogeneous system of linear equations, AX = 0 (B = 0)
1. If the |𝐴| ≠ 0 then the system is consistent with trivial solution x = 0, y = 0, z =
0
2. If |𝐴| = 0 (A is singular) and adjA .B ≠ 0 then the solution does not exist. The
system is inconsistent.
3. If |𝐴| = 0 (A is singular) and adjA .B = 0 then the system is consistent with
infinitely many solutions. to find these solutions put z = k in two of the equations and
solve them by matrix method.