May 2011 - Career Point
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May 2011 - Career Point

May 2011 - Career Point May 2011 - Career Point

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MATHSMATRICES &DETERMINANTSMathematics FundamentalsMatrices :An m × n matrix is a rectangular array of mn numbers(real or complex) arranged in an ordered set of mhorizontal lines called rows and n vertical lines calledcolumns enclosed in parentheses. An m × n matrix Ais usually written as :⎡a11⎢⎢a21⎢ MA = ⎢⎢ ai1⎢ M⎢⎢⎣am1aaaa1222Mi2Mm2............aaa1 j2 jaijmj............a1n⎤a⎥2n⎥⎥⎥ain⎥⎥⎥amn⎥⎦Where 1 ≤ i ≤ m and 1 ≤ j ≤ nand is written in compact form as A = [a ij ] m× nA matrix A = [a ij ] m × n is called(i) a rectangular matrix if m ≠ n(ii) a square matrix if m = n(iii) a row matrix or row vector if m = 1(iv) a column matrix or column vector if n = 1(v) a null matrix if a ij = 0 for all i, j and is denoted byO m× n(vi) a diagonal matrix if a ij = 0 for i ≠ j(vii) a scalar matrix if a ij = 0 for i ≠ j and all diagonalelements a ii are equalTwo matrices can be added only when thye are of sameorder. If A = [a ij ] m × n and B = [b ij ] m × n , then sum of Aand B is denoted by A + B and is a matrix [a ij + b ij ] m × nThe product of two matrices A and B, written as AB,is defined in this very order of matrices if number ofcolumns of A (pre factor) is equal to the number ofrows of B (post factor). If AB is defined , we say thatA and B are conformable for multiplication in theorder AB.If A = [a ij ] m × n and B = [b ij ] n × p , then their product ABis a matrix C = [c ij ] m × p whereC ij = sum of the products of elements of ith row of Awith the corresponding elements of j th column of B.Types of matrices :(i) Idempotent if A 2 = A(ii) Periodic if A k+1 = A for some positive integer k.The least value of k is called the period of A.(iii) Nilpotent if A k = O when k is a positive integer.Least value of k is called the index of thenilpotent matrix.(iv) Involutary if A 2 = I.The matrix obtained from a matrix A = [a ij ] m × n bychanging its rows into columns and columns of A intorows is called the transpose of A and is denoted by A′.A square matrix a = [a ij ] n × n is said to be(i) Symmetric if a ij = a ji for all i and j i.e. if A′ = A.(ii) Skew-symmetric ifa ij = – a ji for all i and j i.e., if A′ = –A.Every square matrix A can be uniquely written as sumof a symmetric and a skew-symmetric matrix.A = 21 (A + A′) + 21 (A – A′) where 21 (A + A′) issymmetric and 21 (A – A′) is skew-symmetric.Let A = [a ij ] m × n be a given matrix. Then the matrixobtained from A by replacing all the elements by theirconjugate complex is called the conjugate of the matrixA and is denoted by A = a ] .Properties :(i) ( A ) = A(ii) ( A + B)= A + B[ ij(iii) (λ A)= λ A , where λ is a scalar(iv) ( A B) = A B .Determinant :Consider the set of linear equations a 1 x + b 1 y = 0 anda 2 x + b 2 y = 0, where on eliminating x and y we getthe eliminant a 1 b 2 – a 2 b 1 = 0; or symbolically, wewrite in the determinant notationa1b1≡ a 1 b 2 – a 2 b 1 = 0a2b2Here the scalar a 1 b 2 – a 2 b 1 is said to be the expansiona1b1of the 2 × 2 order determinant having 2a2b2rows and 2 columns.Similarly, a determinant of 3 × 3 order can beexpanded as :XtraEdge for IIT-JEE 44 MAY 2011

aaa123bbb123ccc123b2c2= a 1b3c3a2c2a2b2– b 1 + c 1a3c3a3b3= a 1 (b 2 c 3 – b 3 c 2 ) – b 1 (a 2 c 3 – a 3 c 2 ) + c 1 (a 2 b 3 – a 3 b 2 )= a 1 (b 2 c 3 – b 3 c 2 ) – a 2 (b 1 c 3 – b 3 c 1 ) + a 3 (b 1 c 2 – b 2 c 1 )= Σ(± a i b j c k )To every square matrix A = [a ij ] m × n is associated anumber of function called the determinant of A and isdenoted by | A | or det A.Thus, | A | =aaa1121Mn1aaa1222Mn2.........aaa1n2nIf A = [a ij ] n × n , then the matrix obtained from A afterdeleting ith row and jth column is called a submatrixof A. The determinant of this submatrix is called aminor or a ij .Sum of products of elements of a row (or column) ina det with their corresponding cofactors is equal tothe value of the determinant.ni.e., ∑i=1a ijC ij = | A | and ∑j=n1Mnna ijC ij = | A |.(i) If all the elements of any two rows or two columnsof a determinant ate either identical orproportional, then the determinant is zero.(ii) If A is a square matrix of order n, then| kA | = k n | A |.(iii) If ∆ is determinant of order n and ∆′ is thedeterminant obtained from ∆ by replacing theelements by the corresponding cofactors, then∆′ = ∆ n–1(iv) Determinant of a skew-symmetric matrix of oddorder is always zero.The determinant of a square matrix can be evaluatedby expanding from any row or column.If A = [a ij ] n × n is a square matrix and C ij is thecofactor of a ij in A, then the transpose of the matrixobtained from A after replacing each element by thecorresponding cofactor is called the adjoint of A andis denoted by adj. A.Thus, adj. A = [C ij ]′.Properties of adjoint of a square matrix(i) If A is a square matrix of order n, thenA . (adj. A) = (adj . A) A = | A | I n .(ii) If | A | = 0, then A (adj. A) = (adj. A) A = O.(iii) | adj . A | = | A | n –1 if | A | ≠ 0(iv) adj. (AB) = (adj. B) (adj. A).(v) adj. (adj. A) = | A | n – 2 A.Let A be a square matrix of order n. Then the inverse ofA is given by A –1 1= adj. A.| A |Reversal law : If A, B, C are invertible matrices of sameorder, then(i) (AB) –1 = B –1 A –1(ii) (ABC) –1 = C –1 B –1 A –1Criterion of consistency of a system of linear equations(i) The non-homogeneous system AX = B, B ≠ 0 hasunique solution if | A | ≠ 0 and the unique solution isgiven by X = A –1 B.(ii) Cramer’s Rule : If | A | ≠ 0 and X = (x 1 , x 2 ,..., x n )′| Athen for each i =1, 2, 3, …, n ; x i =i |where| A |Ai is the matrix obtained from A by replacing theith column with B.(iii) If | A | = 0 and (adj. A) B = O, then the systemAX = B is consistent and has infinitely manysolutions.(iv) If | A | = 0 and (adj. A) B ≠ O, then the systemAX = B is inconsistent.(v) If | A | ≠ 0 then the homogeneous system AX = Ohas only null solution or trivial solution(i.e., x 1 = 0, x 2 = 0, …. x n = 0)(vi) If | A | = 0, then the system AX = O has non-nullsolution.(i) Area of a triangle having vertices at (x 1 , y 1 ), (x 2 , y 2 )and (x 3 , y 3 ) is given by12(ii) Three points A(x 1 , y 1 ), B(x 2 , y 2 ) and C(x 3 , y 3 ) arecollinear iff area of ∆ABC = 0.A square matrix A is called an orthogonal matrix ifAA′ = AA′ = I.A square matrix A is called unitary if AA θ = A θ A = I(i) The determinant of a unitary matrix is of modulusunity.(ii) If A is a unitary matrix then A′, A , A θ , A –1 areunitary.(iii) Product of two unitary matrices is unitary.Differentiation of Determinants :Let A = | C 1 C 2 C 3 | is a determinant thendA = | C′1 C 2 C 3 | + | C 1 C′ 2 C 3 | + | C 1 C 2 C′ 3 |dxSame process we have for row.Thus, to differentiate a determinant, we differentiate onecolumn (or row) at a time, keeping others unchanged.xxx123yyy123111XtraEdge for IIT-JEE 45 MAY 2011

MATHSMATRICES &DETERMINANTSMathematics FundamentalsMatrices :An m × n matrix is a rectangular array of mn numbers(real or complex) arranged in an ordered set of mhorizontal lines called rows and n vertical lines calledcolumns enclosed in parentheses. An m × n matrix Ais usually written as :⎡a11⎢⎢a21⎢ MA = ⎢⎢ ai1⎢ M⎢⎢⎣am1aaaa1222Mi2Mm2............aaa1 j2 jaijmj............a1n⎤a⎥2n⎥⎥⎥ain⎥⎥⎥amn⎥⎦Where 1 ≤ i ≤ m and 1 ≤ j ≤ nand is written in compact form as A = [a ij ] m× nA matrix A = [a ij ] m × n is called(i) a rectangular matrix if m ≠ n(ii) a square matrix if m = n(iii) a row matrix or row vector if m = 1(iv) a column matrix or column vector if n = 1(v) a null matrix if a ij = 0 for all i, j and is denoted byO m× n(vi) a diagonal matrix if a ij = 0 for i ≠ j(vii) a scalar matrix if a ij = 0 for i ≠ j and all diagonalelements a ii are equalTwo matrices can be added only when thye are of sameorder. If A = [a ij ] m × n and B = [b ij ] m × n , then sum of Aand B is denoted by A + B and is a matrix [a ij + b ij ] m × nThe product of two matrices A and B, written as AB,is defined in this very order of matrices if number ofcolumns of A (pre factor) is equal to the number ofrows of B (post factor). If AB is defined , we say thatA and B are conformable for multiplication in theorder AB.If A = [a ij ] m × n and B = [b ij ] n × p , then their product ABis a matrix C = [c ij ] m × p whereC ij = sum of the products of elements of ith row of Awith the corresponding elements of j th column of B.Types of matrices :(i) Idempotent if A 2 = A(ii) Periodic if A k+1 = A for some positive integer k.The least value of k is called the period of A.(iii) Nilpotent if A k = O when k is a positive integer.Least value of k is called the index of thenilpotent matrix.(iv) Involutary if A 2 = I.The matrix obtained from a matrix A = [a ij ] m × n bychanging its rows into columns and columns of A intorows is called the transpose of A and is denoted by A′.A square matrix a = [a ij ] n × n is said to be(i) Symmetric if a ij = a ji for all i and j i.e. if A′ = A.(ii) Skew-symmetric ifa ij = – a ji for all i and j i.e., if A′ = –A.Every square matrix A can be uniquely written as sumof a symmetric and a skew-symmetric matrix.A = 21 (A + A′) + 21 (A – A′) where 21 (A + A′) issymmetric and 21 (A – A′) is skew-symmetric.Let A = [a ij ] m × n be a given matrix. Then the matrixobtained from A by replacing all the elements by theirconjugate complex is called the conjugate of the matrixA and is denoted by A = a ] .Properties :(i) ( A ) = A(ii) ( A + B)= A + B[ ij(iii) (λ A)= λ A , where λ is a scalar(iv) ( A B) = A B .Determinant :Consider the set of linear equations a 1 x + b 1 y = 0 anda 2 x + b 2 y = 0, where on eliminating x and y we getthe eliminant a 1 b 2 – a 2 b 1 = 0; or symbolically, wewrite in the determinant notationa1b1≡ a 1 b 2 – a 2 b 1 = 0a2b2Here the scalar a 1 b 2 – a 2 b 1 is said to be the expansiona1b1of the 2 × 2 order determinant having 2a2b2rows and 2 columns.Similarly, a determinant of 3 × 3 order can beexpanded as :XtraEdge for IIT-JEE 44 MAY <strong>2011</strong>

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