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Note:
In a square matrix,
(a) The pair of elements a_{ij} & a_{ji} are called Conjugate Elements.
Example: a_{12 }and a_{21}(b) The elements a_{11}, a_{22}, a_{33}, ...... a_{nn} are called Diagonal Elements. The line along which the diagonal elements lie is called the "Principal or Leading" diagonal.
(c) Sum of all the diagonal elements, i.e. Σ a_{ii} is known as trace of the matrix. It is denoted as t_{r}(A).
Note:
Min. number of zeros in a diagonal matrix of order n = n(n – 1)
Equality of MatricesA = [a_{ij}] & B = [b_{ij}] will be equal, only if
 Both have the same order
 a_{i j} = b_{i j } for each pair of i & j
For the Matrices,
Note:
 If A and B are two nonzero matrices such that AB = O then A and B are called the divisors of zero.
 If A 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 For a unit matrix I of any order , I^{m} = I for all m ∈ N.
Illustration 1. Find the value of x and y, if
Ans.
∴
On equating the corresponding elements of L.H.S. and R.H.S.
A2 + y = 5 ⇒ y = 3
2x + 2 = 8 ⇒ 2x = 6 ⇒ x = 3
Thus, x = 3 and y = 3.
Illustration 2. If , find AB and BA if possible.
Ans. A is a 3 × 3 matrix and B is a 3 × 2 matrix, therefore, A and B are conformable for the product AB and it is of the order 3 × 2.
BA is not possible since the number of columns of B ≠ number of rows of A.
(a) Idempotent Matrix: A square matrix is idempotent provided A^{2} = A
(b) Nilpotent Matrix: A square matrix is said to be nilpotent matrix of order m, m ∈ N , if A^{m} = O, A^{m–1} ≠ O
(c) Periodic Matrix: A square matrix is periodic when it satisfies the relation A^{K+1} = A, for some positive integer K. The period of the matrix is the least value of K for which this holds true.
(d) Involutary Matrix: If A^{2} = I , the matrix is said to be an involutary matrix.
Note:
 For an idempotent matrix, A^{n} = A ∀ n > 2, n ∈ N.
 Period of an idempotent matrix is 1.
 For an involuntary Matrix, A = A^{–1}.
Properties of Transpose:
If A^{T} & B^{T }denote the transpose of A and B.
(a) (A ± B)^{T} = A^{T} ± B^{T} ; note that A & B have the same order.
(b) (A B)^{T} = B^{T} A^{T} A & B are conformable for matrix product AB.
(c) (A^{T})^{T} = A
(d) (kA)^{T} = k A^{T} k is a scalar.
Note:
(i) For symmetric matrix, A = A^{T}
(ii) Max. number of distinct entries in a symmetric matrix of order n is
(iii) The digaonal elements of a skew symmetric matrix are all zero, but not the converse i.e. if digaonal elements are 0 doesn't mean matrix is skew symmetric.
(a) A is symmetric, if A^{T} = A and A is skewsymmetric, if A^{T} = − A.
(b) A + A^{T} is a symmetric matrix and A − A^{T} is a skew symmetric matrix.
Consider (A + A^{T})^{T} = A^{T} + (A^{T})^{T} = A^{T} + A = A + A^{T}. Thus, A + A^{T} is symmetric.
Similarly, we can prove that A − A^{T }is skewsymmetric.
(c) The sum of two symmetric matrices is a symmetric matrix and the sum of two skewsymmetric matrices is a skewsymmetric matrix.
(d) If A & B are symmetric matrices then,
(AB + BA) is a symmetric matrix and (AB − BA) is a skewsymmetric matrix.
(e) Every square matrix can be uniquely expressed as a sum of a symmetric and a skewsymmetric matrix.
i.e.
Let be a square matrix and let the matrix formed by the cofactors of [a_{i j} ] in determinant A is = Then (adj A) =
Note: Cofactors of the elements of any matrix are obtained by eliminating all the elements of the same row and column and calculating the determinant of the remaining elements.
Theorem : A (adj. A) = (adj. A).A = A I_{n}, if A be a square matrix of order n.
Properties:
(i)  adj A  =  A ^{n – 1}
(ii) adj (AB) = (adj B) (adj A)
(iii) adj(KA) = K^{n–1} (adj A), K is a scalar
Illustration 3. If , find adj A.
Ans. Each element of cofactor matrix
Cofactor matrix =
adj A = transpose of codaftor matrix =
Any one of the following operations on a matrix is called an elementary transformation.
Imp. Theorem: If A & B are invertible matrices of the same order , then (AB)^{−1} = B^{−1}A^{−1}. This is reversal law for inverse
Note:
 The necessary and sufficient condition for a square matrix A to be invertible is that A ≠ 0.
 If A be an invertible matrix, then A^{T} is also invertible & (A^{T})^{−1} = (A^{−1})^{T}.
 If A is invertible,
(a) (A^{−1})^{−1} = A ;
(b) (A^{k})^{−1} = (A^{−1})^{k} = A^{–k}, k ∈ N If A is an Orthogonal Matrix. AA^{T} = I = A^{T}A
 A square matrix is said to be orthogonal if , A^{−1} = A^{T}.
a1x + b1y + c1z = d1
a2x + b2y + c2z = d2
a3x + b3y + c3z = d3
We can write it in the form of matrix:
i.e.
Let
⇒ AX = B
⇒ A^{−1} A X = A^{−1} B
⇒ X = A^{−1} B =
Note:
 If A ≠ 0, system is consistent having unique solution
 If A ≠ 0 & (adj A). B ≠ O (Null matrix), system is consistent having unique non − trivial solution.
 If A ≠ 0 & (adj A) . B = O (Null matrix) , system is consistent having trivial solution
 If A = 0, matrix method fails
Illustration 4. Solve the following equation,
2x+y+2z=0, 2xy+z=10, x+3yz=5.
Ans. The given equation can be written in matrix form:
∵X = A^{−1} B =
Therefore,
⇒
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209 videos218 docs139 tests
