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Matrices Matrices Presentation Transcript

  • Presented by Ajay Gupta
  • AJAY GUPTA PGT MATHSCONT. NO. 9868423152 KV VIKASPURI, NEW DELHI
  • MATRICES A matrix is a rectangular array (arrangement) of numbers real or imaginary or functions kept inside braces () or [ ]subject to certain rules of operations. 3 2 3 4 5  3   2 4 5 
  • ORDER OF A MATRIX A matrix having ‘m’ number of rows and ‘n’ number of columns is said to be of order ‘m ×n’I row 1 − 1 1 II row 2 1 − 3III row   1  1 1  I II III Columns
  • Notation of a Matrix1. In compact form matrix is represented by A = [a i j ] m × n2. The element at i th row and j th column is called the (i, j) th element of the matrix i.e. in a i j the first subscript i always denotes the number of row and j denotes the number of column in which the element occur.3. A matrix having 2 rows and 3 columns is of order 2 × 3 and another matrix having 1 row and 2 columns is of order 1 × 2.
  • Location of the elements in a matrix For matrix A a11 a12 a13  a a 22  a 23   21 a31  a 32 a 33   − 2 5 6 − 7 9 7 4 3  6    5
  • TYPES Of MATRICES MATRICES ROW MATRIX COLUMN MATRIX SQUARE MATRIX SQUARE MATRIX DIAGONAL MATRIX ZERO MATRIX SCALAR MATRIX SYMMTRIC MATRIX IDNTITY MATRIXSKEW-SYMMETRIC MATRIX
  • ROW / COLUMN MATRICES1. Matrix having only one row is called Row- Matrix i.e. the row matrix is of order 1 × n.2. Matrix having only one column is called Column- [2 5 8] matrix i.e. the column matrix is of order m × 1.  − 5 4  
  • ZERO MATRIXA matrix whose all the elements are zero is called zero matrix or null matrix and is denoted by O i.e. a i j = 0 for all i, j. 0  0 0  0  0 0     
  • 1. SQUARE matrix is a matrix having same number of rows and columns and square matrix having ‘n’ number of rows and columns is called of order n2. DIAGONAL matrix is a square matrix if all its elements except in leading diagonal are zero i. e. a ij = 0 for i ≠ j and a ij ≠ 0 for i = j.3. SCALAR matrix is the diagonal matrix with all the elements in leading diagonal matrix are same i.e. a ij = 0 for i ≠ j. and a ij = k for i = j.4. UNIT matrix is the scalar matrix with all the elements in leading diagonal 1 i.e. a ij = 0 for i ≠ j. and a ij = 1 for i = j.
  • 1 4 1 2 − 5 6 1 0   2 0 8 9   8 9  0 −1 0 2  7  2 3      2 0 00 − 2 0 1 0 0 0 1 0   0 1  1 2    1 0    0 0 3  0 0 1      2 1 1 0 0   0 1 2  1 0 0 0 1 0   0 0 1  0 − 1 0        0 0 1   2 4 0      0 0 1   
  • OPERATION ON MATRICES Matrices support different basic operations .Some of the basic operations that can be applied are1. Addition of matrices.2. Subtraction of matrices.3. Multiplication of matrices.4. Multiplication of matrix with scalar value.But two matrices can not be divided.
  • EQUALITY OF MATRICESTwo matrices are EQUAL if both are of same order and each of the corresponding element in both the matrices is same. 1 2  1 3 5  1 − 2  1 − 2 4 3 4 3 43   2 4 6    5 6    − 7 8    − 7 − 8    [2 5 9] [2 5 9]
  • ADDITION OF MATRICESTwo or more matrices of same order can be add up to form single matrix of same order. 2 − 1 4  8 0 3 − 7 5 6 +  − 1 2 4   
  • ADDITION OF MATRICESTwo or more matrices of same order can be add up to form single matrix of same order. 2 − 1 4  8 0 3 − 7 5 6 +  − 1 2 4    2 + 8 _ _  _  _ _ 
  • ADDITION OF MATRICESTwo or more matrices of same order can be add up to form single matrix of same order. 2 − 1 4 8 0 3 − 7 5 6 + − 1  2 4   2 + 8 − 1 + 0 _  _ _  _ 
  • ADDITION OF MATRICESTwo or more matrices of same order can be add up to form single matrix of same order. 2 − 1 4 8 0 3 − 7 5 6 + − 1 2  4    2 + 8 − 1 + 0 4 + 3  _ _ _  
  • ADDITION OF MATRICESTwo or more matrices of same order can be add up to form single matrix of same order. 2 − 1 4  8 0 3 − 7 5 6  +  − 1 2 4     2 + 8 − 1 + 0 4 + 3 − 7 − 1 _ _  
  • ADDITION OF MATRICESTwo or more matrices of same order can be add up to form single matrix of same order. 2 − 1 4  8 0 3 − 7 5 6  +  − 1 2 4     2 + 8 − 1 + 0 4 + 3 − 7 − 1 5 + 2 _  
  • ADDITION OF MATRICESTwo or more matrices of same order can be add up to form single matrix of same order. 2 − 1 4  8 0 3 − 7 5 6  +  − 1 2 4     2 + 8 − 1 + 0 4 + 3  − 7 − 1 5 + 2 6 + 4  
  • ADDITION OF MATRICES Two or more matrices of same order can be add up to form single matrix of same order. 2 − 1 4 8 0 3 − 7 5 6 + − 1  2 4    2 + 8 − 1 + 0 4 + 3  10 − 1 7 = =  − 8 7 10  − 7 − 1 5 + 2 6 + 4  
  • PROPERTIES OF MATRIX ADDITION A+B=B+A A + ( B + C) = (A + B) + CA+ 0 = 0 +A=A A + (-A) = 0 = (-A) + AA+ B =A+ C  B = C
  • MULTIPLICATION OF MATRIX WITH SCALAR For matrix A of order m × n and scalar number k, the matrix of order m × n obtained by multiplying each element of A with k is called scalar multiplication of A by k and is denoted by kA. 1 − 2 3 For A =  4 5 0   1× 2 _ _ 2A =  _ _  _ 
  • MULTIPLICATION OF MATRIX WITH SCALAR For matrix A of order m × n and scalar number k, the matrix of order m × n obtained by multiplying each element of A with k is called scalar multiplication of A by k and is denoted by kA. 1 − 2 3 For A = 4 5  0  2 _ _ 2A = _ _  _ 
  • MULTIPLICATION OF MATRIX WITH SCALAR For matrix A of order m × n and scalar number k, the matrix of order m × n obtained by multiplying each element of A with k is called scalar multiplication of A by k and is denoted by kA. 1 − 2 3 For A =  4 5 0    2 2 × −2 _  2A = _ _ _ 
  • MULTIPLICATION OF MATRIX WITH SCALAR For matrix A of order m × n and scalar number k, the matrix of order m × n obtained by multiplying each element of A with k is called scalar multiplication of A by k and is denoted by kA. 1 − 2 3 For A =  4 5 0   2 − 4 _ 2A = _ _ _   
  • MULTIPLICATION OF MATRIX WITH SCALAR For matrix A of order m × n and scalar number k, the matrix of order m × n obtained by multiplying each element of A with k is called scalar multiplication of A by k and is denoted by kA. 1 − 2 3 For A =  4 5 0    2 − 4 2 × 3 2A =   _ _ _ 
  • MULTIPLICATION OF MATRIX WITH SCALAR For matrix A of order m × n and scalar number k, the matrix of order m × n obtained by multiplying each element of A with k is called scalar multiplication of A by k and is denoted by kA. 1 − 2 3 For A =    4 5 0 2 − 4 6 2A =   _ _ _ 
  • MULTIPLICATION OF MATRIX WITH SCALAR For matrix A of order m × n and scalar number k, the matrix of order m × n obtained by multiplying each element of A with k is called scalar multiplication of A by k and is denoted by kA. 1 − 2 3 For A =  4 5 0    2 − 4 6 2A = 2 × 4 _ _  
  • MULTIPLICATION OF MATRIX WITH SCALAR For matrix A of order m × n and scalar number k, the matrix of order m × n obtained by multiplying each element of A with k is called scalar multiplication of A by k and is denoted by kA. 1 − 2 3 For A =   4 5 0 2 − 4 6 2A =   8 _ _
  • MULTIPLICATION OF MATRIX WITH SCALAR For matrix A of order m × n and scalar number k, the matrix of order m × n obtained by multiplying each element of A with k is called scalar multiplication of A by k and is denoted by kA. 1 − 2 3 For A =  4 5 0   2 − 4 6  2A = 8 2 × 5 _   
  • MULTIPLICATION OF MATRIX WITH SCALAR For matrix A of order m × n and scalar number k, the matrix of order m × n obtained by multiplying each element of A with k is called scalar multiplication of A by k and is denoted by kA. 1 − 2 3 For A =  4 5 0   2 −4 6 2A = 8 10  _ 
  • MULTIPLICATION OF MATRIX WITH SCALAR For matrix A of order m × n and scalar number k, the matrix of order m × n obtained by multiplying each element of A with k is called scalar multiplication of A by k and is denoted by kA. 1 − 2 3 For A =  4 5 0   2 − 4 6  2A = 8 10 2 × 0  
  • MULTIPLICATION OF MATRIX WITH SCALAR For matrix A of order m × n and scalar number k, the matrix of order m × n obtained by multiplying each element of A with k is called scalar multiplication of A by k and is denoted by kA. 1 − 2 3 For A =  4 5 0   2 −4 6 2A = 8 10  0 
  • MULTIPLICATION OF MATRIX WITH SCALAR For matrix A of order m × n and scalar number k, the matrix of order m × n obtained by multiplying each element of A with k is called scalar multiplication of A by k and is denoted by kA. 1 − 2 3 For A =   4 5 0 − 3 _ _ -3A =   _ _ _
  • MULTIPLICATION OF MATRIX WITH SCALAR For matrix A of order m × n and scalar number k, the matrix of order m × n obtained by multiplying each element of A with k is called scalar multiplication of A by k and is denoted by kA. 1 − 2 3 For A =  4 5 0   − 3 6 _ -3A = _ _  _ 
  • MULTIPLICATION OF MATRIX WITH SCALAR For matrix A of order m × n and scalar number k, the matrix of order m × n obtained by multiplying each element of A with k is called scalar multiplication of A by k and is denoted by kA. 1 − 2 3 For A =  4 5 0    − 3 2 − 9 -3A =  _ _ _  
  • MULTIPLICATION OF MATRIX WITH SCALAR For matrix A of order m × n and scalar number k, the matrix of order m × n obtained by multiplying each element of A with k is called scalar multiplication of A by k and is denoted by kA. 1 − 2 3 For A =  4 5 0    − 3 2 − 3 -3A = − 12 _ _   
  • MULTIPLICATION OF MATRIX WITH SCALAR For matrix A of order m × n and scalar number k, the matrix of order m × n obtained by multiplying each element of A with k is called scalar multiplication of A by k and is denoted by kA. 1 − 2 3 For A =  4 5 0    −3 2 − 3 -3A =  − 12 − 15 _   
  • MULTIPLICATION OF MATRIX WITH SCALAR For matrix A of order m × n and scalar number k, the matrix of order m × n obtained by multiplying each element of A with k is called scalar multiplication of A by k and is denoted by kA. 1 − 2 3 For A =  4 5 0    − 3 2 − 3 -3A =  − 12 − 15 0   
  • PROPERTIES OF SCALAR MULTIPLICATION k (A + B) = k A + k B (-k) A = - (k A) = k (-A) IA=AI=A (-1) A = - A
  • MULTIPLICATION OF MATRICES Two matrices can be multiplied only if number of columns of first is same as number of rows of the second. If A is of order m × n and B is of order n × p, then the product AB is a matrix of order m × p. m × n & n × p  m × p. For A = [a i j] m×n and B = [ b j k] n×p , AB = C with C = [cij] m×p where ci k = Σ a ij b jk
  • MULTIPLICATION OF MATRICES 6 9   2 6 0   2 3  7 8 9 = 
  • MULTIPLICATION OF MATRICES 6 9  2 6 0  6 .2 + 9 .7 _ _   =    7 8 9    2 3    _ _ _
  • MULTIPLICATION OF MATRICES 6 9  2 6 0  6.2 + 9.7 6.6 + 9.8 _    =     7 8 9  _ _ _  2 3  
  • MULTIPLICATION OF MATRICES 6 9   2 6 0   6.2 + 9.7 6.6 + 9.8 6.0 + 9.9   7 8 9 =        2 3    _ _ _  
  • MULTIPLICATION OF MATRICES 6 9  2 6 0  6.2 + 9.7 6.6 + 9.8 6.0 + 9.9      7 8 9 =   2.2 + 3.7    2 3    _ _ 
  • MULTIPLICATION OF MATRICES 6 9  2 6 0  6. 2 + 9. 7 6. 6 + 9. 8 6. 0 + 9. 9      7 8 9 =   2.2 + 3.7 2.6 + 3.8 _    2 3    
  • MULTIPLICATION OF MATRICES 6 9  2 6 0  6.2 + 9.7 6.6 + 9.8 6.0 + 9.9     7 8 9  =   2.2 + 3.7 2.6 + 3.8 2.0 + 3.9    2 3    
  • MULTIPLICATION OF MATRICES6 9   2 6 0   6.2 + 9.7 6.6 + 9.9 6.0 + 9.8   ×   =   2 3   7 9 8   2.2 + 3.7 2.6 + 3.9 2.0 + 3.8 
  • MULTIPLICATION OF MATRICES 6.2 + 9.7 6.6 + 9.9 6.0 + 9.8  =  2.2 + 3.7 2.6 + 3.9 2.0 + 3.8    75 117 72  =   25 30    24 
  • TRANPOSE OF MATRIX For matrix A = [aij] of order m×n, / transpose of A is denoted by A of A and T it is a matrix of order n×m and is obtained by interchanging the rows with columns i.e. AT=[aji] with aij = aji for all i,j.
  •  1 5  9 8 A  1 9 − 1A=   T =   − 1 3 3× 2  5 8 3  2× 3  
  • PROPERTIES OFTRANSPOSE OF MATRICES (AT)T = A (A + B)T= AT + BT (kA)T = k AT (AB)T = BT AT Every square matrix can be expressed as sum of sum of symmetric and skew-symmetric 1 (A + AT) + 1(A – AT) matrix. A = 2 2
  • SYMMETRIC/SKEW-SYMMETRIC MATRICES A square matrix A = [aij] is called symmetric matrix if AT = A i.e. aij = aji for all i,j. A square matrix A = [aij] is called skew- symmetric matrix if AT = -A i.e. aij = - aji for all i,j.  1 2 − 3  0 −2 3  2 0 7 2  0 − 7     − 3 7 − 2 − 3 7  0  
  • IMPORTANT RESULT ON SYMMETRICAND SKEW-SYMMETRIC MATRICES Everysquare matrix can be expressed as sum of sum of symmetric and skew- symmetric matrix. A =(A + AT) +(A – AT) Allthe elements in lead diagonal in skew- symmetric matrix are zero.
  • APPLICATION OF MATRICES Solution of equations in AX=B system using matrix method −1 (i) If A = 0 unique solution with X = A B / (ii) If A = 0, and also (adjA)B = 0, Infinite many solutions. (iii) If A = 0, (adj A) B = 0 No solution. /
  • Important Problems1 Construct a 2 3 matrix A with elements given by i +2 j aij = i−j2 Find x, y such that  x − y 2 − 2 3 − 2 2  6 0 0  4  + 1 0 − 1 = 5 2 x + y 5 x 6      3 If A = diag.(2 -5 9), B = diag.(1 1 -4), find 3A – 2B. 3 2 1 04 Find X and Y if 2X + Y = 1 4 and X + 2Y =     − 3 2 
  • 1 3 2   15 Find x if [ 1 x 1]  2 5 1   0 2  =  15  3 2   x  3 16 If A =  − 1 2 show that A 2 − 5 A + 7 = 0  .  1    w w2   w   w2  1 1       w 2 w 1 + w 2  1 w  w  =07 Show that    w2  1 w  1   w 2    w w  2     3 18 If A = − 1 2 and Find K so that A 2 = 5 A + KI   .9 Show that B ′AB is symmetric or skew- symmetric according as A is symmetric or skew-symmetric.
  • 10 Express A = 3 2 as sum of symmetric 3 4 5 3    2 4 5  and skew-symmetric matrices.11 If A = 3 2 find ( AB ) 4 6 −1 2 5  & B −1 =     3 2 12 Find X if 3 2  − 1 1  2 − 1 7 5 X − 2 1 = − 1 4       13 Solve using matrix method x + 2y + z = 7, x + 3 z = 11, 2 x – 3 y = 1.
  • Address of the subject related websites  http://www.netsoc.tcd.ie/~jgilbert/maths_site/apple  http://www.ping.be/~ping1339/matr.htm  http://mathworld.wolfram.com/Matrix.html  http://en.wikipedia.org/wiki/Matrices
  • ACKNOWLEDGEMENT This power point presentation is prepared under the active guidance of Ms. Summy and Ms. Nidhi the able and learned trainers of project “SHIKSHA” CONDUCTED BY MICROSOFT CORPORATION.