DETERMINANT AND MATRICS
Determinant : A determinant is shown by D. If we have a set of linear equations l1x + m1y = 0 and 12x + m2y = 0. On eliminating x and y we have l1m2 – l2m1 = 0. It can be written as
l1m1
= (l m – l m ) = 0
12m2
1 2 2 1
It is a 2 C 2 order determinant.
A 3 C 3 order determinant can be written as
l1m1t1
l m t =1
m2t2 –
2 2 2
1 m3t3
l3m3t3
m l2t2 +t
1 l3t3 1
l2m2 l3m3
odd.
= l1 (m2t3 – m3t2) – m1 l2t3 – t2l3) + t1(m2l3 – l2m3)
The sign is taken as positive or negative according as permutation is even or
l1m1t1 In a determinant l2m2t2
l3m3t3
= l1x1 – my1 + t1z1 = l1x1 -12×2 + 13×3 Here x1, y1, z1, (x1, x2, x3) are called the minors
of the elements l1, m1,t1 (l1,l2,l3) respectively. Thus
x = m2t2 ,x = m1t1 ,x = m1t1
1 m3t3 2 m3t3 3 m2t2
Co-factors of a Determinant : In a
x11x12x1
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determinant D = x
x22
x23
x31x32x33
The co-factor C
of a = (-1)i+j M , where M
is the minor of a . For example,
C12
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= (-1)1+2M
ij ij
= -M12
ij ij ij
= – x21x22
x31x33
This can be made clear by considering
1 2 3
determinant 3 2 1 – Minor of the
2 3 1
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element in 1st row and 1st element i.e. 1= 2 1 = 2 – 3 = -1, 2 = 3 1 = 3 – 2 = 1
3 = 3 2 = 9 – 4 = 5 cofactor of 1 = (-1)1+1 (-1) = -1
2 3
Cofactor of a 2 = (-1)1 + 2 (1) = – 1
Properties of a Determinant :
- If all the elements of a row or column are zero, the determinant value is also
zero.
- On changing the rows or columns of a determinant the value of the determi-
nant remains the same.
- If the elements of a row or column are identical to the elements of any other row or column then the value of determinant becomes zero.
- The interchange of any two adjacent columns of a determinant changes its sign
- On multiplying all the elements of a row or column by a constant k, then the value of the determinant becomes K times.
- If the elements of a row or column are shown as sum or difference of two quantities, then the determinant can be shown as the sum or difference of two determi- nants with the same order, e.,
x1 + x2 y1 + y2 z1 + z2
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D = a b c
a3 b3 c3
x1y1z1 x2y2z2
= a2b2c2 + a2b2c2 a3b3c3 a3b3c3
- If some multiple of elements of any row or column are added or subtracted from the corresponding elements of any other row or column then the determinant remains unchanged, e.,
If the operations R ® R + mR + nR , j, k ¹ i and C ® C + mC + nC , j, k ¹ i are
i i i k i i j k
performed on the determinant, it remains unchanged.
- If a determinant D = f(x) where f(x) is a polynomial in x, vanishes when x = 1, then (x – 1) is factor of D.
Multiplication of two Determinants : The product of two determinants is ob- tained as follows :
a1b1c1 x1y1z1
a2b2c2 x x2y2z2
a3b3c3 x3y3z3
a1x1 + b1y1 + c1z1, a1x2 + b1y2 + c1z2 a1x3 + b1y3 + c1z3
= a2x1 + b2y1 + c2z1, a2x2 + b2y2 + c2z2 a2x3 + b2y3 + c2z2 a3x1 + b3y1 + c3z1, a3x2 + b3y2 + c3z2 a3x3 + b3y3 + c3z3
Here the rows have been multiplied by rows. Rows can also be multiplied by columns.
Differentiation of a Determinant : Let D(x) be a determinant of order two. If we
write D(x) = |x x |, where x
and x
denote the 1st and 2nd columns, then
1
D(x) = |x¢
2 1 2
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x2 + x1 x¢ |,
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where x ¢j’ denotes the column which contains the derivative of all the functions in the jth column x.
In a similar fashion, if we write D(x)
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= R1 then D¢(x) = R ¢ + R1 .
viz.
2 R2 R2¢
If only one row or column consists functions of x and other rows are constant,
f1(x) f2(x) f3(x)
Let D(x) = l1 l2 13 say
m1 m2 m3
f1¢(x) f2¢(x) f3¢(x)
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then D¢(x) = l l l
2 3
m1 m2 m3 and in general Dn(x)
f n(x) f n(x) f n(x)
1 2 3
= l1 l2 l3
m1 m2 m3
where n is any positive integer and fn(x) shows the nth derivative of f(x). In general, if D is n C n determinant, D1 = Dn-1.
Cramer’s Rule : Let D be the determinant of coefficient of the simultaneous linear equations l1x + m1y + t1z = d1, l2x + m2y + t2z = d2 and l3x + m3y + t3z = d3, then
l1 m1 t1
D = l2 m2 t2 l3 m3 t3
x x x x
If D ¹ 0, then —- = —- = —- =—– i.e.,
D1 D2 D3 D
x x
= = ———
d1 m1 t1 l1 d1 t1
d2 m2 t2 l2 d2 t2
d3 m3 t3 l3 d3 t3
z l
= = ———
l1 m1 d1 l1 m1 t1
l2 m2 d2 l2 m2 t2
l3 m3 d3 l3 m3 t3
If D = 0, then Cramer’s rule cannot be applied. We have similar Cramer’s rule for
simultaneous linear equations l1x + m1y = d1 and l2x + m2y = d2.
The system of following homogeneous equations l1x + m1y + t1z = 0, l2x + m2y + t2z = 0, l3x + m3y + t3z = 0 is always consistent.
l1 m1 d1
If D = l2 m2 d2 then this has the l3 m3 d3
unique solution x = y = z = 0 known as trivial solution. But if D = 0, then this has
an infinite number of solutions. Hence for non-trivial solution D = 0.
The System of Linear Equations :
l1x + m1y + t1z = d1 l1 m1 t1 l2x + m2y + t2z = d2 with D = l2 m2 t2 l3x + m3y + t3z = d3 l3 m3 t3
is consistent (with unique solution) if D x y z 1
¹ 0 and —- = —- = —- = —-.
D1 D2 D3 D
Inconsistent (with no solution) if D = 0 and at least one of the D1, D2 and D3 is
non-zero.
Consistent (with infinite solution) if D = D1 = D2 = D3 = 0.
Degree of the determinant : The degree of a determinant is decided by the
degree of the expression obtained by multiplying the elements of the main diagonal.
Matrices : A matrix refers to a rectangular arrangement of certain number (real or complex) or of certain function. If a system of m linear equations in n unknowns is given as
a11y1 + a12y2 +…. + a1nyn = l1
a21y1 + a22y2 +…. + a2nyn = l2
………………………………………..
am1yn + am2yn + + amnyn = lm
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a11 a22 … a1n y1 l1 a21 a22 … a2n y2 l2
… … … … … …
am1 am2 amn yn ln
a11 a12……. a1n
a21 a22……. a2n
where A =…………………… is called
am1am2……. amn
an m C n matrix with elements aij (i = 1, 2, …, m; j = 1, 2,…………………………………………………………… , n) over the field of
real (complex) numbers.
If a matrix is having m rows and n columns, it has a order m C n.
TYPES OF MATRICES
Square matrix : If a matrix has the number of rows equal to the number of col- umns it is called a square matrix.
Identity matrix : A square matrix which has all non-diagonal elements as zero and all the diagonal elements as unity is called a unit matrix or an identity matrix.
1 0 0
Thus 0 1 0 and
0 0 1
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 0
are unit matrix, denoted by 1.
Zero matrix or Null matrix : Any m C n matrix which has all the elements as
zero is called a zero matrix or null matrix of the type m C n and is denoted by Om C n.
0 0
Thus 0 0 is 0 matrix.
0 0 3C2
Row Matrix : A 1 C n matrix with only one row is called a row matrix. e.g., a = [a11 a12 … a1n]l C n. or a = [1 2 3]
Column matrix : A n — 1 matrix with only one column is called a column
a11
a21 1
matrix e.g.a = … or a = 2
am1 m C 1 3
Numerically written as the transpose of a row matrix a = [a11 a21………………………………………………………………………………………………………………. am1].
Upper triangular matrix : In a square matrix in which i > j is known as upper
triangular matrix.
a b c a = 0 d e
0 0 f 3 C 3
Lower triangular matrix : A square matrix in which aij = 0, i < j, is known as a lower triangular matrix.
a 0 0 e.g.,a = b c 0
d e f 3 C 3
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Diagonal matrix : A matrix, whose all elements are zero except those d
…) in the leading diagonal is called
a 0 0 a diaggnal matrix e.g. 0 b 0
0 0 c
(i = 1, 2,
Scalar matrix : A diagonal matrix having c as a scalar is called a scalar matrix.
All its leading diagonal elements are equal to c.
Symmetric matrix : A square matrix A is said to be symmetric if a¢ = a.
Skew-Symmetric matrix : A square matrix A is said to be skew-symmetric if a¢ =
The diagonal elements of the skew. symmetric matrix are all zero.
Note : Every square matrix can be represented as the sum of a symmetric and a
skew-symmetric matrix.
Hermitian matrix : A square matrix is said to be Hermitian matrix, if its conjugate transpose is equal to the matrix itself. i.e., a = (aij)n + n is Hermitian if a, = a (i.e., if aij = aij
“ i and j).
Skew-Hermitian matrix : a = (aij)nCn is said to be skew-Hermitian if a, = – a (i.e., aij = aij, “ and j).
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Nilpotent matrix : A square matrix a = (a ) is said to be nilpotent (of order 2)
if a2 = 0.
A square matrix is said to be Nilpotent of order p if p is the least positive integer such that ap = 0.
Idempotent matrix : A square matrix a = (aij)n C n
is said to be idempotent if A2
1 0
= a. e.g., 0 1 is an idempotent matrix.
Involutory matrix : A square matrix a = (aij)n C n is said to be involutory if a =
2
1 0
l, l being the identity matrix e.g., 0 1 is an involutory matrix.
Orthogonal matrix : A square matrix a = (aij)n C n is called an orthogonal matrix if a¢ a = 1 = a¢ a.
Equal matrices : Two matrices a and b are said to be equal, written as a = b, if
- both are of the same order e., have the same number of rows and columns,
and
- The elements in the corresponding places of the two matrices are the
Negative of a matrix : If a be a given matrix then -a is called its negative matrix.
In fact all its elements are the corresponding elements of A multiplied by -1.
Scalar multiple of a matrix : If A is a matrix whose all elements are made K times of the corresponding elements of a.
l[a ] = [Ka ] .
ij mCn ij mCn
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(k + l) a + ka + la (distributive law) (kl) a = k (la) = l (ka) (associative law)
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ij mCn
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ij mCn
ij = bij
[aij]mCn + [bij]mCn = [aij + bij]mCn
a + b = b + a (commutative law)
(a + b) + c + a + (b + c) (associative law) k(a + b) = ka + kb (distributive law)
a + O = a = O + a
a + (-a) = O = (-a) + a
a – b = a + (- b)
Matrix Multiplication : If a = [aij]mCp¢ b = [bij]pCn and c = ab = [Cij]mCn, where p
Cij = å aij bjk i.e., cik = a11 b1k + a12b2k + … + aip bpk. j=1
(ab) c = (ab) c (associative law)
a(b + c) = ab + ac (b + c) a = ba + ca
(distributive law)
(ka) b = k (ab) = a(kb)
Ia = a = aI, I Î R (or C) where a is a square matrix and I is the unit matrix of the same order.
Oa = O = aO, O Î R (or C) where a is a square matrix and O is the null matrix of the same order.
If a is m C n matrix, then Im a = a = aIn If ab = 0, then a or b need not be equal to zero matrix.
Transpose of a Matrix : Suppose A is a m C n matrix. Then the matrix of order n
C m obtained by changing its rows into columns and columns into rows is called that transpose of a and is denoted by a¢ or aT or aI.
(a + b + c + … + I)¢ = a¢ + b¢ + c¢ + … + I–
(ab … I)¢ = I¢ … b¢a¢
(ka)¢ = ka¢ (a¢)¢ = a
Conjugate of a Matrix : If a is a matrix having complex numbers as its elements, the matrix obtained from a by replacing each element of a by its conjugate is called the conjugate of a and is denoted by a.
Properties of conjugates :
a = a
a + b = a + b
(K a) = K a, (K being any complex number) (ab) = ab
(a¢) = (a¢) = aq
l1 m1 t1 l1 m1 t1
a = l2 m2 t2 then a = l2 m2 t2
l3 m3 t3 l3 m3 t3 3C3
|ab| = |a| |b| = |b| |a|
a is singular if and only if |a| = 0 a is non-singular or invertible if and only if
|a| ¹ 0.
Let a = [aij]nCn be a square matrix and Cij the cofactor of aij in |a|, then the trans- pose of the matrix b obtained by replacing each element of a by its cofactor in |a| is known as the adjoint of a and is denoted by adj b.
i.e., adj a = b¢, where b = [Cij] a(adj a) = (adj a) a = |a| 1
adj (ab) = (adj b) (adj a)
If a is a square matrix of the order n, then adj (adj a) = |a|n – 2 a; and det (adj a) = (det a)n – 1 if |a|¹ 0, then
1
a-1 =—– C (adj) a
|a|
aa-1 = a-1 a = 1 (a-1)-1 = a
(ab)-1 = b-1a-1
(a …a )-1 = a -1…a -1…a -1a -1
1 n n 2 2 1
(a¢)-1 = (a-1)i
1
(ka)-1 = — a-1 if k ¹ 0
k
If a
a b
= c d
and |a| ¹ 0, then a-1
d -b
ad – bc ad – bc
= c a
ad – bc ad – bc
Consistency of a System of Linear Equation ax = b, where a is a n – Square Matrix : In a system of linear equations ax = b, a = (aij)n C n is said to
- Consistent (with unique solution) if |a| ¹ 0, e., if a is non-singular matrix.
- Inconsistent (it has no solution) if |a| = 0 and (adj a) b is a non-null matrix.
- Consistent (with infinitely many solutions) if |a| = 0 and (adj a) b is a null
matrix.
Some Properties of Matrices and Determinants :
- det b = k det a, if b is obtained from a multiplying one row or column of a by k.
- det (ka) = kn det (a) where a is a matrix of order n C Hence det (ka) = k3 det
- where a is a matrix of order 3 C
- If a,b, c are square matrices of the same order that 1th column (or rows) of b and c and all other columns (or rows) of a, b and c are identical then det a = det b + det
- a = a+ is always symmetric
- a – a¢ is a skew-symmetric
- aa¢ is a symmetric
- The matrix bT ab is symmetric or skew-symmetric according as a is symmetric or skew-symmetric.
- All positive integral powers of a symmetric matrix are
- Positive odd integral powers of a skew-symmetric matrix are skew-symmetric and positive even integral powers of a skew-symmetric matrix are
- The inverse of a symmetric matrix is symmetric.
- The inverse of a diagonal matrix is
- A diagonal matrix is both an upper triangular and a lower triangular.
- If s is a scalar matrix and a is a square matrix of same order, then as =
- Suppose a and b are two symmetric matrices of order n, then aba is a sym- metric
- Suppose a is a square matrix of odd order, then |-a| = – |a|.
- Suppose a is a non-singular
1 1
matrix, then det (a-1) = ——— i.e., |a-1| = —–.
det (a) |a|
- All orthogonal matrices are
- Each invertible matrices is not necessarily
- Each skew-symmetric matrices of odd order is not
- The multiplication of two diagonal matrices of the same order is also a diago- nal
- The adjoint of a diagonal matrix is also a diagonal matrix.
- If a is an orthogonal matrix, then a-1 is also
- Suppose a and b are m-rowed square matrices such that they commute, then
(a + b)n = c nan + c nan-1b + n an-2 b2+…..+n bn
o l c2 cn
- Determinant of a skew-symmetric matrix of odd order is zero and of even order is a non-zero.
- If a = b + c, then it is not necessary that det a = det b + det
- The determinant of a diagonal matrix = product of its diagonal elements.
- The determinant of a diagonal matrix = product of its diagonal elements.
- The determinant value of a unity matrix is
- The determinant of a null matrix is
