May 2011 - Career Point

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