Matrices
1. Definition: Rectangular array of mn numbers. Unlike determinants, it has no value.
\(A=\left[ \begin{array}{cccc}{a_{11}} & {a_{12}} & {\dots \ldots} & {a_{1 n}} \\ {a_{21}} & {a_{22}} & {\dots \ldots} & {a_{2 n}} \\ {\vdots} & {\vdots} & {\vdots} & {\vdots} \\ {a_{m 1}} & {a_{m 2}} & {\dots \ldots} & {a_{m n}}\end{array}\right]\text {Or}\left( \begin{array}{cccc}{a_{11}} & {a_{12}} & {\dots \dots} & {a_{1 n}} \\ {a_{21}} & {a_{22}} & {\dots \ldots} & {a_{2 n}} \\ {\vdots} & {\vdots} & {\vdots} & {\vdots} \\ {a_{m 1}} & {a_{m 2}} & {\dots \dots} & {a_{m n}}\end{array}\right)\)
Abbreviated as: A = [ aij ] 1 ≤ i ≤ m ; 1 ≤ j ≤ n, i denotes the row and j denotes the column is called a matrix of order m × n.
2. Special Type Of Matrices:
- Row Matrix: A = [ a11 , a12 , …… a1n ] having one row . (1 × n) matrix. (or row vectors)
- Column Matrix:
\(A=\left[ \begin{array}{c}{a_{11}} \\ {a_{21}} \\ {\vdots} \\ {a_{m 1}}\end{array}\right]\)
having one column. (m × 1) matrix (or column vectors) - Zero or Null Matrix: (A = Om×n ) An m × n matrix all whose entries are zero .
\(A=\left[ \begin{array}{ll}{0} & {0} \\ {0} & {0} \\ {0} & {0}\end{array}\right]\)
is a 3 × 2 null matrix &
\(B=\left[ \begin{array}{lll}{0} & {0} & {0} \\ {0} & {0} & {0} \\ {0} & {0} & {0}\end{array}\right]\)
is 3 × 3 null matrix - Horizontal Matrix: A matrix of order m × n is a horizontal matrix if n > m.
\(\left[ \begin{array}{llll}{1} & {2} & {3} & {4} \\ {2} & {5} & {1} & {1}\end{array}\right]\) - Verical Matrix: A matrix of order m × n is a vertical matrix if m > n.
\(\left[ \begin{array}{ll}{2} & {5} \\ {1} & {1} \\ {3} & {6} \\ {2} & {4}\end{array}\right]\) - Square Matrix: (Order n)If number of row = number of column ⇒ a square matrix.
Note:
(i) In a square matrix the pair of elements aij & aji are called Conjugate Elements .e.g.
\(\left( \begin{array}{ll}{a_{11}} & {a_{12}} \\ {a_{21}} & {a_{22}}\end{array}\right)\)
(ii) The elements a11 , a22 , a33 , …… ann are called Diagonal Elements . The line along which the diagonal elements lie is called “Principal or Leading” diagonal. The qty Σ aii = trace of the matrice written as , i.e. tr A Triangular Matrix Diagonal Matrix denote as ddia (d1 , d2 , ….., dn) all elements except the leading diagonal are zero diagonal Matrix Unit or Identity Matrix
Note: Min. number of zeros in a diagonal matrix of order n = n(n – 1) “It is to be noted that with square matrix there is a corresponding determinant formed by the elements of A in the same order.”
3. Equality Of Matrices:
Let A = [aij ] & B = [bij ] are equal if ,
- both have the same order.
- aij = b1ij for each pair of i & j.
4.Algebra Of Matrices:
Addition: A + B = [ aij + bij ] where A & B are of the same type. (order)
- Addition of matrices is commutative. i.e. A + B = B + A, A = m × n; B = m × n
- Matrix addition is associative .(A + B) + C = A + (B + C) Note : A , B & C are of the same type.
- Additive inverse. If A + B = O = B + A A = m × n
5. Multiplication Of A Matrix By A Scalar:
\(\text { If A }=\left[ \begin{array}{lll} { { a } } & { { b } } & { { c } } \\ { { b } } & { { c } } & { { a } } \\ { { c } } & { { a } } & { { b } } \end{array} \right];\) \(\mathrm{k} \mathrm{A}=\left[ \begin{array}{lll}{\mathrm{ka}} & {\mathrm{kb}} & {\mathrm{kc}} \\ {\mathrm{kb}} & {\mathrm{kc}} & {\mathrm{ka}} \\ {\mathrm{kc}} & {\mathrm{ka}} & {\mathrm{kb}}\end{array}\right]\)
6.Multiplication Of Matrices: (Row by Column)AB exists if, A = m × n & B = n × p 2 × 3 3 × 3
AB exists , but BA does not ⇒ AB ≠ BA
Note: In the product AB,
\(\left\{\begin{array}{l}{\mathrm{A}=\text { pre factor }} \\ {\mathrm{B}=\text { post factor }}\end{array}\right.\)
A = (a1 , a2 , …… an) &
\(B=\left[ \begin{array}{c}{b_{1}} \\ {b_{2}} \\ {\vdots} \\ {b_{n}}\end{array}\right]\)
1 × n n × 1 A B = [a1b1 + a1b2 + …… + anbn]
If A = [ aij ] m × n & B = [ bij] n × p matrix , then
\((\mathrm{AB})_{\mathrm{ij}}=\sum_{\mathrm{r}=1}^{\mathrm{n}} \mathrm{a}_{\mathrm{ir}} \cdot \mathrm{b}_{\mathrm{rj}}\)
Properties Of Matrix Multiplication:
- Matrix multiplication is not commutative.
\(A =\left[ \begin{array}{ll}{1} & {1} \\ {0} & {0}\end{array}\right];\) \(B=\left[ \begin{array}{ll}{1} & {0} \\ {0} & {0}\end{array}\right];\) \(AB=\left[ \begin{array}{ll}{1} & {0} \\ {0} & {0}\end{array}\right];\) \(\mathrm{BA}=\left[ \begin{array}{ll}{1} & {1} \\ {0} & {0}\end{array}\right] \Rightarrow \mathrm{AB} \neq \mathrm{BA}(\text { in general })\) - \(A B=\left[ \begin{array}{ll}{1} & {1} \\ {2} & {2}\end{array}\right] \left[ \begin{array}{cc}{-1} & {1} \\ {1} & {-1}\end{array}\right]=\left[ \begin{array}{ll}{0} & {0} \\ {0} & {0}\end{array}\right]\)
⇒ AB = O ⇒/ A = O or B = O
Note: IfA and B are two non- zero matrices such that AB = O then A and B are called the divisors of zero. Also if [AB] = O ⇒ |AB| ⇒ |A||B| = 0 ⇒ |A| = 0 or |B| = 0 but not the converse. IfA and B are two matrices such that
(i) AB = BA ⇒ A and B commute each other
(ii) AB = – BA ⇒ A and B anti commute each other
3. Matrix Multiplication Is Associative:
If A , B & C are conformable for the product AB & BC, then (A . B) . C = A . (B . C)
4. Distributivity:
\(\left.\begin{aligned} \mathrm{A}(\mathrm{B}+\mathrm{C}) &=\mathrm{AB}+\mathrm{AC} \\(\mathrm{A}+\mathrm{B}) \mathrm{C} &=\mathrm{AC}+\mathrm{BC} \end{aligned}\right]\)
Provided A, B & C are conformable for respective products
5. Positive Integral Powers Of A Square Matrix:
For a square matrix A , A² A = (A A) A = A (A A) = A3.
Note that for a unit matrix I of any order, Im = I for all m ∈ N.
6. Matrix Polynomial:
If f (x) = a0xn + a1xn-1 + a2xn-2 + ……… + anx0 then we define a matrix polynomial f(A) = a0An + a1An-1 + a2An-2 + ….. + anIn where A is the given square matrix. If f (A) is the null matrix then A is called the zero or root of the polynomial f (x).
Definitions:
- Idempotent Matrix: A square matrix is idempotent provided A2 = A.
Note that An = A ∀ n ≥ 2, n ∈ N. - Nilpotent Matrix: A square matrix is said to be nilpotent matrix of order m, m ∈ N, if Am = O , Am-1 ≠ O.
- Periodic Matrix: A square matrix is which satisfies the relation Ak+1 = A, for some positive integer K, is a periodic matrix. The period of the matrix is the least value of K for which this holds true.
Note that period of an idempotent matrix is 1. - Involutary Matrix: IfA2 = I, the matrix is said to be an involutary matrix.
Note that A = A-1 for an involutary matrix.
7. The Transpose Of A Matrix: (Changing rows & columns)
Let A be any matrix. Then, A = aij of order m × n
⇒ AT or A′ = [ aij ] for 1 ≤ i ≤ n & 1 ≤ j ≤ m of order n × m
Properties of Transpose of a Matrix:
If AT & BT denote the transpose of A and B ,
- (A ± B)T = AT ± BT ; note that A & B have the same order.
- (AB)T = BTAT A & B are conformable for matrix product AB.
- (AT)T = A
- (kA)T = kAT k is a scalar.
- General : (A1, A2, …… An)T = AnT , ……. , A2T , A1T (reversal law for transpose)
8. Symmetric & Skew Symmetric Matrix:
A square matrix A = [ aij] is said to be , symmetric if, aij = aji ∀ i & j (conjugate elements are equal)
(Note A = AT)
Note: Max. number of distinct entries in a symmetric matrix of order n is \(\frac {n(n+1)}{2}\)
and skew symmetric if , aij = − aji ∀ i & j (the pair of conjugate elements are additive inverse of each other) (Note A = –AT ) Hence If A is skew symmetric, then aii = − aii ⇒ aii = 0 ∀ i Thus the digaonal elements of a skew symmetric matrix are all zero , but not the converse.
Properties Of Symmetric & Skew Matrix:
- Property 1: A is symmetric if AT = A
s skew symmetric if AT = − A - Property 2: A + ATis a symmetric matrix A − AT is a skew symmetric matrix. Consider (A + AT)T = AT + (AT)T = AT + A = A + AT
A + AT is symmetric.
Similarly, we can prove that A − AT is skew symmetric. - Property 3: The sum of two symmetric matrix is a symmetric matrix and the sum of two skew symmetric matrix is a skew symmetric matrix. Let AT = A; BT = B where A & B have the same order. (A + B)T = A + B Similarly we can prove the other.
- Property 4: If A & B are symmetric matrices then,
(a) AB + BA is a symmetric matrix
(b) AB − BA is a skew symmetric matrix . - Property 5: Every square matrix can be uniquely expressed as a sum of a symmetric and a skew symmetric matrix.
\(\begin{array}{rl}{\mathrm{A}=\frac{1}{2}\left(\mathrm{A}+\mathrm{A}^{\mathrm{T}}\right)+} & {\frac{1}{2}\left(\mathrm{A}-\mathrm{A}^{\mathrm{T}}\right)} \\ {\mathrm{P}} & {\mathrm{Q}} \\ {\text { Symmetric }} & {\text { Skew Symmetric }}\end{array}\)
9. Adjoint Of A Square Matrix:
Let
\(A=\left[a_{i j}\right]=\left( \begin{array}{lll}{a_{11}} & {a_{12}} & {a_{13}} \\ {a_{21}} & {a_{22}} & {a_{23}} \\ {a_{31}} & {a_{32}} & {a_{33}}\end{array}\right)\)
be a square matrix and let the matrix formed by the cofactors of [aij ] in determinant
\(|\mathrm{A}| \text { is }=\left( \begin{array}{lll}{\mathrm{C}_{11}} & {\mathrm{C}_{12}} & {\mathrm{C}_{13}} \\ {\mathrm{C}_{21}} & {\mathrm{C}_{22}} & {\mathrm{C}_{23}} \\ {\mathrm{C}_{31}} & {\mathrm{C}_{32}} & {\mathrm{C}_{33}}\end{array}\right)\)
Then,
\((\text {adj A})=\left( \begin{array}{ccc}{\mathrm{C}_{11}} & {\mathrm{C}_{21}} & {\mathrm{C}_{31}} \\ {\mathrm{C}_{12}} & {\mathrm{C}_{22}} & {\mathrm{C}_{32}} \\ {\mathrm{C}_{13}} & {\mathrm{C}_{23}} & {\mathrm{C}_{33}}\end{array}\right)\)
Theorem: A (adj. A) = (adj. A).A = |A| In , If A be a square matrix of order n.
Note: If A and B are non singular square matrices of same order, then
(i) |adj A| = |A|n-1
(ii) adj (AB) = (adj B) (adj A)
(iii) adj(KA) = Kn-1 (adj A), K is a scalar
Inverse Of A Matrix (Reciprocal Matrix): A square matrix A said to be invertible (non singular) if there exists a matrix B such that, AB = I = BA
B is called the inverse (reciprocal) of A and is denoted by A-1. Thus
A-1 = B ⇔ AB = I = BA . We have, A . (adj A) = |A| In
A-1 A (adj A) = A-1I|Α|;
In (adj A) = A-1 |A| In
∴ \( A^{-1} = \frac {(adj A)}{|A|}\)
Note: The necessary and sufficient condition for a square matrix A to be invertible is that |A| ≠ 0.
Theorem: IfA & B are invertible matrices ofthe same order , then (AB)-1 = B-1 A-1 . This is reversal law for inverse
Note: (i)If A be an invertible matrix , then AT is also invertible & (AT)-1 = (A-1)T .
(ii) If A is invertible, (a) (A −1) −1 = A ; (b) (Ak)-1 = (A-1)k = A-k, k ∈ N
(iii) IfA is an Orthogonal Matrix. AAT= I = AT
(iv) A square matrix is said to be orthogonal if , A-1= AT.
\((v)|A^{-1}|=\frac {1}{|A|}\)
System Of Equation & Criterian For Consistency
Gauss – Jordan Method
x + y + z = 6, x − y + z = 2, 2 x + y − z = 1
or
\(\left( \begin{array}{c}{x+y+z} \\ {x-y+z} \\ {2 x+y-z}\end{array}\right)=\left( \begin{array}{l}{6} \\ {2} \\ {1}\end{array}\right) \quad \left( \begin{array}{ccc}{1} & {1} & {1} \\ {1} & {-1} & {1} \\ {2} & {1} & {-1}\end{array}\right) \left( \begin{array}{l}{x} \\ {y} \\ {z}\end{array}\right)=\left( \begin{array}{l}{6} \\ {2} \\ {1}\end{array}\right)\)
A X = B ⇒ A −1 A X = A −1 B ⇒
\(\mathrm{X}=\mathrm{A}^{-1} \mathrm{B}=\frac{(\text { adj. } \mathrm{A}) . \mathrm{B}}{|\mathrm{A}|}\)
Note:
(1)If |A| ≠ 0, system is consistent having unique solution
(2)If |A| ≠ 0 & (adj A) . B ≠ O (Null matrix) , system is consistent having unique non − trivial solution.
(3) If |A| ≠ 0 & (adj A) . B = O (Null matrix) , system is consistent having trivial solution
(4) If