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1 DGT MH –CET 12th MATHEMATICS Study Material Matrices 18 02 Matrices iv. Square Matrix : Amatrix in which number of Syllabus rows is equal to the number of columns, is called a square matrix. The elements a of a Types of Matrices Algebra of Matrices ij square matrix A = [a ) m × m for which i = j Equality of Two Matrices Trace of a Matrix. ij i.e. the elements a , a ··· a are called the Equivalent Matrix Inverse of a Matrix. 11 22 mm Applications of Matrices diagonal elements and the line along which called the principal diagonal or leading diagonal In Mathematics, a matrix (plural matrices) is a of the matrix; rectangular array of numbers, symbols or 1 2 3 expressions, arranged in rows and columns. The 3 2 1 individualsin a matrix are called its elements or e.g.A = is a square matrix of order 2 3 1 entries. Generally, matrix is written in the following 33 way : in which diagonal elements are 1,2, l. v. Null Matrix or Zero Matrix A matrix of order m × n whose all elements are zero is called a a a ... a null matrix of order m × n. 11 12 1n It is denoted by 0. a21 a22... a2n 0 0 0 0 0 A = = [a ] m × n e.g. 0 = and ij 0 0 0 0 0 am1 a amn m2 are two null matrices of order 2×2 and 2 ×3, where, a is the entry at ith row andjth column. respectively. ij The orderofamatrixAismx n,wheremis the number vi. Diagonal Matrix A square matrix is called a of rows and n is the number of columns. diagonal matrix, if all its non-diagonal elements Types of Matrices are zero and diagonal elements mayor may i. Row Matrix : A matrix which has only one not be zero. row and any nuymber of columns, is called a If d , d , d ......,d are elements of principal row matrix. 1 2 3 n diagonal of a diagonal matrix of order n x n, e.g. A = [27 85 1 4] is a row matrix. then matrix is denoted as diag [d , d ,...... d ] 1 × 4 1 2 n ii. Column Matrix Amatrix is said to be a column a 0 0 matrix, if it has only one column and any number of rows. e.g. A = 0 b 0 is a diagonal matrix which is 0 0 c 1 2 a diagonal matrix which is denoted by A = e.g. A = is a column matrix. diag [a, b, c]. 3 31 Note : The number of zeroes in a diagonal matrix iii. Rectangular Matrix Amatrix in which number lie between n2 – n to n2, where n is an order of of rows is not equal to the number of columns the matrix. or vice-versa is called a rectangular matrix. vii: Triple-Diagonal Matrix A square matrix A is 1 2 3 said to be a triple-diagonal matrix, if all its e.g. A = is a rectangular matrix of elements are zero except possibly for those 4 5 6 lying on the principal diagonal, the diagonal order 2 × 3. immediately above as well as below the principal diagonal. MATHEMATICS – MHT-CET Himalaya Publication Pvt. Ltd. DGT Group - Tuitions (Feed Concepts) XIth – XIIth | JEE | CET | NEET | Call : 9920154035 / 8169861448 2 DGT MH –CET 12th MATHEMATICS Study Material Matrices 19 1 1 0 0 matrix A (not all) is known as sub matrix of A i.e. 5 0 the matrix B constituted by the array of elements, 1 2 1 0 which are left after deleting some rows or columns 3 4 3 or both of matrix A is called submatrix of A. e.g. A = and 0 1 2 3 (a) Principal Submatrix A square submatrix B 0 0 4 0 0 4 5 of a square matrix A is called a principal submatrix, viii. Scalar Matrix. A square matrix A = [aij Iis if the diagonal elements of B are also diagonal said to be scalar matrix, if elements of A. (a) a = 0, 0,i j (b) Leading Submatrix A principal square ij ij submatrix B is said to be a leading submatrix of a (b)a = 0, i h, wherek 0 square matrix A, if it is obtained by deleting only ij some of the last rows and the corresponding In other words, a diagonal matrix is said to be a scalar matrix, if the elements of principal diagonal columns such that the leading elements (i.e. au) is are same. not lost 5 0 0 xiii. Horizontal Matrix Any matrix in which the number of columns is more than the number of e.g. A = 0 5 0 is a scalar matrix. rows is called a horizontal matrix. 0 0 5 2 3 4 5 ix. Limit Matrix or Identity Matrix 8 9 7 2 A square matrix A = [a is said to be a unit matrix e.g. is a horizontal matrix. ijl 2 2 3 4 or identity matrix, if (a) a = 0, i j xiv.Vertical Matrix Any matrix in which the ij number of rows is more than the number of (b) a = 1, i j ij columns is called column matrix. In other words, A diagonal matrix, whose elements 2 3 of principal diagonal are equal to 1 and all remaining elements are zero, is known as unit or 4 5 identity matrix. It is denoted by 1. e.g. is a column matrix. 6 7 32 1 0 0 Algebra of Matrices 0 1 0 Four types of algebra of matrices are defined e.g. I = is a unit matrix of order 3. below: 0 0 1 1. Addition of Two Matrices x. Upper Triangular Matrix Let A = [a ] and B = [b ] are two matrices A square matrix A = [a is known as upper ij m×n ij m×n triangular matrix, if ij whose orders are same, then A + B = [a + b ] i1,2....,mand j1,2,....n a = 0, i j ij ij ij Example 1 0 1 0 2 3 5 1 e.g. A = 0 1 0 is an upper triangular matrix. If A = 0 3 0 and 0 0 1 xi. Lower Triangular Matrix A square matrix 2 5 1 A = [a ] is known as triangular matrix, if [a = 0 B = 2 3 1/2 then A + B is ij ij i j 1 0 0 2 3 5 1 4 2 0 a. 0 3 0 e.g. A = is a lower triangular matrix. 5 6 3 xii. Submatrices of a Matrix A matrix B obtained b. 3 1 5 1 by deleting the row (s) or column (s) or both of a 2 6 1/2 MATHEMATICS – MHT-CET Himalaya Publication Pvt. Ltd. DGT Group - Tuitions (Feed Concepts) XIth – XIIth | JEE | CET | NEET | Call : 9920154035 / 8169861448 3 DGT MH –CET 12th MATHEMATICS Study Material Matrices 20 2 3 1 5 0 1 7 3 1 c. c. d. 0 6 1/2 5 7 5 3 2 4 1 3 2 3 1 5 1 Sol (b) A – B = – d. 3 2 2 5 0 6 1/2 Sol (c) Since, A and B are of the same order 2 × 3. 21 43 1 1 = = Therefore, addition of A and B is defined and is 3(2) 25 5 3 given by 3. Scalar Multiplication Let A = [a ] be any m×n matrix and k be any ij 2 3 1 5 11 scalar. Then, the matrix obtained by multiplying A + B = 1 each element of A by k is called the scalar 22 33 0 multiplication of A by k and it is denoted by kA. 2 Thus, if A = [a ] , then kA = [ka ] ij m×n ij m×n 2 3 1 5 0 1 2 3 2 4 6 = 1 3 2 1 6 4 2 0 6 e.g. If A = , then 2A = 2 1 3 1 2 6 2 Properties of Addition of Matrices Properties of Scalar Multiplication Let A, Band C are three matrices of same order, If A = [a ] and B = [b ] are two matrices then ij m×n ij m x n i. Matrix addition is commutative and , are two scalars, then i.e.A + B = B + A i. (A + B) = A + AB ii. Matrix addition is associative, ii. (A + ) A = A + A i.e. (A + B) + C = A + (B + C) iii. (+ ) A = A (A) = ( A) iii. If 0 is a null matrix of order m × n and iv. (–) A = – (A) = A(–) A + 0 =A =0 + A, then 0 is known as additive 4. Multiplication of Two Matrices If A = [a m × n and B = [b ] are two matrices identity. ij ij m×n iv. If for each matrix A = [a ] a matrix (–A) is such that the number of columns of A is equal to such that ij m×n the number of rows of B, then a matrix C = [c ] of order m x p is known as product A + (– A) = 0 = (–A) + A, ij m × p then matrix (– A) is known as additive inverse of of matrices A and B, where A . c n ij = a b j b a b ...a b v. Matrix addition follows cancellation law, ik k 1j i2 2j in nj i.e. A + H = A + C B = C (left cancellation law) k1 and it is denoted by C = AB. and B+A=C+ A B = C (right cancellation law) Transpose of a Matrix Note Two matrices are said to be conformable If A = [a ] is a matrix of order m × n, then the for addition or subtraction. if they are of the same ij m×n order. transpose of A can be obtained by changing all 2. Subtraction of Two Matrices rows to columns and all columns to rows i.e. transpose of A =[a ]n × m Let A = [a ] and B = [b ] are two matrices Tji ij m×n ij m×n It is denoted by A', A or At. of same order. Then, A – B = C = [C ] , ij mxn 1 4 where c = a – b , ij ij ij 1 2 3 Example 2 2 5 T e.g. Let A = , then AA = 4 5 6 3 6 23 2 4 1 3 32 If A = and B = , then A – B is 3 2 2 5 Properties of Transpose of a Matrix 3 7 1 1 If A and B are two matrices and k is a scalar, then a. b. i. (A')' = A ii. (A + B)' = A' + B' 5 7 5 3 iii: (kA)' = kA' iv.(AB)'=B'A' (reversal law) MATHEMATICS – MHT-CET Himalaya Publication Pvt. Ltd. DGT Group - Tuitions (Feed Concepts) XIth – XIIth | JEE | CET | NEET | Call : 9920154035 / 8169861448 4 DGT MH –CET 12th MATHEMATICS Study Material Matrices 21 Note If A. Band C are any three matrices If A and B are idempotent matrices, then A +B is conformable for multiplication. then (ABC)' = an idempotent, if AB = – BA. C'B' A'. ii. Nilpotent Matrix A square matrix A is called Conjugate of a Matrix nilpotent matrix, if it satisfies the relation The matrix obtained from any given matrix A Ak = 0 and Ak–1 0. containing complex numbers as its elements, on where, k is a positive integer. replacing its elements by the corresponding iii. Involutory Matrix A square matrix A is called conjugate complex numbers is called conjugate an involutory matrix, if it satisfies the relation of A and is denoted by A iv. Symmetric Matrix A square matrix A is called symmetric matrix, if it satisfies the relation i2i 23i A'=A e.g. if A = 4 5i 56i, then If A and B are symmetric matrices of the same order, then 12i 23i (a) AB is symmetric if AB = BA. A 45i 56i (b)A ± B, AB + BA are also symmetric matrices. If A is symmetric matrix, then A-I will also be Properties of Conjugate of a Matrix symmetric matrix. If A and B are two matrices, then v. Skew-symmetric Matrix i. (A)A ii. (AB)AB A square matrix A is called skew-symmetric matrix, iii. iv. if it satisfies the relation ABA.B (kA)kA,kisareal scalar A' = – A Conjugate Transpose of a Matrix. If A and B are two skew-symmetric matrices, then The transpose of the conjugate of a matrix A is (a) A + B, AB – BA are skew-symmetric matrices. called conjugate transpose of A and is denoted by (b)AB + BA is a symmetric matrix. A or A. Determinant of skew-symmetric matrix of odd A– = Conjugate of A' = (A) order is zero. Note The transpose of the conjugate of A is the Note Every square matrix can be uniquely same as the conjugate of the transpose of A expressed as the sum of symmetric and skew- symmetric matrix. i.e.A = 1 (A+A') + 1 (A–A') 24i 3 59i 2 2 4 i 3i e.g. If A = 2 5 4i 1 1 where. (A + A') is symmetric and (A – A') 24i 4 2 2 2 is skew-symmetric. 3 i 5 Example 3 then A– = 59i 3i 4i 6 9 2 6 0 Properties of Conjugate Transpose ofa If A = 2 3 and B = 7 9 8, then AB is Matrix 75 25 117 75 117 72 i. For a matrix A, (A') = CAY a. b. ii. (A–)– = A 72 39 24 25 39 24 iii. If A and B are two matrices, then 72 29 24 (A+B)– = A– + B– c. d. Not defined e e 75 25 117 iv. (kA) = kA , where k is any real scalar. – – – v. (AB) = B A Sol (b) The matrix A has 2 columns which is Special Types of Matrices equal to the number of rows of B. Hence, AB is i. Idempotent Matrix A square matrix A is called defined. an idempotent matrix, if it satisfies the relation A2 = A. MATHEMATICS – MHT-CET Himalaya Publication Pvt. Ltd. DGT Group - Tuitions (Feed Concepts) XIth – XIIth | JEE | CET | NEET | Call : 9920154035 / 8169861448
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