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Transcript of Matrices
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Presented by
Ajay Gupta
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AJAY GUPTA PGT MATHSCONT. NO. 9868423152
KV VIKASPURI, NEW DELHI
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MATRICESMATRICES A matrix is a rectangular array A matrix is a rectangular array
(arrangement) of numbers real (arrangement) of numbers real or imaginary or functions kept or imaginary or functions kept inside braces () or [ ]subject to inside braces () or [ ]subject to certain rules of operations.certain rules of operations.
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ORDER OF A MATRIXORDER OF A MATRIX
A matrix having ‘m’ number of A matrix having ‘m’ number of rows rows and ‘n’ number of and ‘n’ number of columnscolumns is said to be is said to be of order ‘m n’of order ‘m n’
I rowI row
II rowII row
III rowIII row
I II IIII II III
ColumnsColumns
111
312
111
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Notation of a MatrixNotation of a Matrix
1.1. In compact form matrix is represented by In compact form matrix is represented by A = [a A = [a i j i j ] m × n
2.2. The element at i The element at i thth row and j row and j thth column is called column is called the (i, j) the (i, j) thth element of the matrix i.e. in element of the matrix i.e. in a a i j i j the first the first subscript i always denotes the number of row subscript i always denotes the number of row and j denotes the number of column in which the and j denotes the number of column in which the element occur.element occur.
3.3. A matrix having 2 rows and 3 columns is of order A matrix having 2 rows and 3 columns is of order 2 2 × 3 and another matrix having 1 row and 2 × 3 and another matrix having 1 row and 2 columns is of order 1 × 2.columns is of order 1 × 2.
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Location of the elements in a matrixLocation of the elements in a matrix
For matrix AFor matrix A
333231
232221
131211
aaa
aaa
aaa
347
652
56
97
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TYPES Of MATRICESTYPES Of MATRICES
MATRICES
ROW MATRIX
COLUMN MATRIX
SQUARE MATRIX
ZERO MATRIX
SYMMTRIC MATRIX
SKEW-SYMMETRIC MATRIX
SQUARE MATRIX
DIAGONAL MATRIX
SCALAR MATRIX
IDNTITY MATRIX
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ROW / COLUMN MATRICESROW / COLUMN MATRICES 1. Matrix having only one row is called Row- 1. Matrix having only one row is called Row-
Matrix i.e. the row matrix is of order 1 Matrix i.e. the row matrix is of order 1 × n.× n.
2. Matrix having only one column is called Column-2. Matrix having only one column is called Column-matrix i.e. the column matrix is of order m × 1.matrix i.e. the column matrix is of order m × 1.
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ZERO MATRIXZERO MATRIX
A matrix whose all the elements are A matrix whose all the elements are zero is called zero matrix or null zero is called zero matrix or null matrix and is denoted by O i.e. amatrix and is denoted by O i.e. ai ji j = = 0 for all i, j.0 for all i, j.
0
0
00
00
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1.1. SQUARESQUARE matrix is a matrix having matrix is a matrix having same number of rows and columns and same number of rows and columns and square matrix having ‘n’ number of square matrix having ‘n’ number of rows and columns is called of order nrows and columns is called of order n
2.2. DIAGONALDIAGONAL matrix is a square matrix if matrix is a square matrix if all its elements except in leading all its elements except in leading diagonal are zero i. e. adiagonal are zero i. e. a ij ij = 0 for i ≠ j = 0 for i ≠ j and aand a ij ij ≠ 0 for i = j. ≠ 0 for i = j.
3.3. SCALARSCALAR matrix is the diagonal matrix matrix is the diagonal matrix with all the elements in leading with all the elements in leading diagonal matrix are same i.e. adiagonal matrix are same i.e. a ij ij = 0 for = 0 for i ≠ j. and ai ≠ j. and a ij ij = k for i = j. = k for i = j.
4.4. UNITUNIT matrix is the scalar matrix with all matrix is the scalar matrix with all the elements in leading diagonal 1 i.e. athe elements in leading diagonal 1 i.e. a
ijij = 0 for i ≠ j. and a = 0 for i ≠ j. and a ij ij = 1 for i = j. = 1 for i = j.
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98
41
327
986
521
300
020
002
10
01
100
010
001
01
10
100
010
001
042
100
210
100
010
001
20
02
12
21
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OPERATION ON MATRICESOPERATION ON MATRICES
Matrices support different basic operationsMatrices support different basic operations..
Some of the basic operations that can be applied areSome of the basic operations that can be applied are
1. Addition of matrices.1. Addition of matrices.
2. Subtraction of matrices.2. Subtraction of matrices.
3. Multiplication of matrices.3. Multiplication of matrices.
4. Multiplication of matrix with scalar value.4. Multiplication of matrix with scalar value.
But two matrices can not be divided.But two matrices can not be divided.
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EQUALITY OF MATRICESEQUALITY OF MATRICES
Two matrices are Two matrices are EQUALEQUAL if both are of same if both are of same order and each of the corresponding element order and each of the corresponding element in both the matrices is same. in both the matrices is same.
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531
87
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87
43
21
952 952
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ADDITION OF MATRICESADDITION OF MATRICES
Two or more matrices of same order can be add up to Two or more matrices of same order can be add up to form single matrix of same order. form single matrix of same order.
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ADDITION OF MATRICESADDITION OF MATRICES
Two or more matrices of same order can be add up to Two or more matrices of same order can be add up to form single matrix of same order. form single matrix of same order.
657
412
421
308
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ADDITION OF MATRICESADDITION OF MATRICES
Two or more matrices of same order can be add up to Two or more matrices of same order can be add up to form single matrix of same order. form single matrix of same order.
657
412
421
308
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_0182
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ADDITION OF MATRICESADDITION OF MATRICES
Two or more matrices of same order can be add up to Two or more matrices of same order can be add up to form single matrix of same order. form single matrix of same order.
657
412
421
308
___
340182
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ADDITION OF MATRICESADDITION OF MATRICES
Two or more matrices of same order can be add up to Two or more matrices of same order can be add up to form single matrix of same order. form single matrix of same order.
657
412
421
308
__17
340182
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ADDITION OF MATRICESADDITION OF MATRICES
Two or more matrices of same order can be add up to Two or more matrices of same order can be add up to form single matrix of same order. form single matrix of same order.
657
412
421
308
_2517
340182
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ADDITION OF MATRICESADDITION OF MATRICES
Two or more matrices of same order can be add up to Two or more matrices of same order can be add up to form single matrix of same order. form single matrix of same order.
657
412
421
308
462517
340182
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ADDITION OF MATRICESADDITION OF MATRICES
Two or more matrices of same order can be add up to Two or more matrices of same order can be add up to form single matrix of same order. form single matrix of same order.
657
412
421
308
1078
7110
462517
340182
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PROPERTIES OF MATRIX PROPERTIES OF MATRIX ADDITIONADDITION
A + B = B + AA + B = B + A A + ( B + C) = (A + B) + CA + ( B + C) = (A + B) + C A + 0 = 0 + A = AA + 0 = 0 + A = A A + (-A) = 0 = (-A) + AA + (-A) = 0 = (-A) + A A + B = A + C A + B = A + C B = C B = C
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MULTIPLICATION OF MATRIX MULTIPLICATION OF MATRIX WITH SCALARWITH SCALAR
For matrix A of order m For matrix A of order m × n × n and scalar number k, the and scalar number k, the matrix of order m matrix of order m × n × n obtained by multiplying each obtained by multiplying each element of A with k is called scalar multiplication of element of A with k is called scalar multiplication of A by k and is denoted by kA.A by k and is denoted by kA.
For A =For A =
2A = 2A =
054
321
___
__21
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MULTIPLICATION OF MATRIX MULTIPLICATION OF MATRIX WITH SCALARWITH SCALAR
For matrix A of order m For matrix A of order m × n × n and scalar number k, the and scalar number k, the matrix of order m matrix of order m × n × n obtained by multiplying each obtained by multiplying each element of A with k is called scalar multiplication of element of A with k is called scalar multiplication of A by k and is denoted by kA.A by k and is denoted by kA.
For A =For A =
2A = 2A =
054
321
___
__2
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MULTIPLICATION OF MATRIX MULTIPLICATION OF MATRIX WITH SCALARWITH SCALAR
For matrix A of order m For matrix A of order m × n × n and scalar number k, the and scalar number k, the matrix of order m matrix of order m × n × n obtained by multiplying each obtained by multiplying each element of A with k is called scalar multiplication of element of A with k is called scalar multiplication of A by k and is denoted by kA.A by k and is denoted by kA.
For A =For A =
2A = 2A =
054
321
___
_222
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MULTIPLICATION OF MATRIX MULTIPLICATION OF MATRIX WITH SCALARWITH SCALAR
For matrix A of order m For matrix A of order m × n × n and scalar number k, the and scalar number k, the matrix of order m matrix of order m × n × n obtained by multiplying each obtained by multiplying each element of A with k is called scalar multiplication of element of A with k is called scalar multiplication of A by k and is denoted by kA.A by k and is denoted by kA.
For A =For A =
2A = 2A =
054
321
___
_42
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MULTIPLICATION OF MATRIX MULTIPLICATION OF MATRIX WITH SCALARWITH SCALAR
For matrix A of order m For matrix A of order m × n × n and scalar number k, the and scalar number k, the matrix of order m matrix of order m × n × n obtained by multiplying each obtained by multiplying each element of A with k is called scalar multiplication of element of A with k is called scalar multiplication of A by k and is denoted by kA.A by k and is denoted by kA.
For A =For A =
2A = 2A =
054
321
___
3242
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MULTIPLICATION OF MATRIX MULTIPLICATION OF MATRIX WITH SCALARWITH SCALAR
For matrix A of order m For matrix A of order m × n × n and scalar number k, the and scalar number k, the matrix of order m matrix of order m × n × n obtained by multiplying each obtained by multiplying each element of A with k is called scalar multiplication of element of A with k is called scalar multiplication of A by k and is denoted by kA.A by k and is denoted by kA.
For A =For A =
2A = 2A =
054
321
___
642
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MULTIPLICATION OF MATRIX MULTIPLICATION OF MATRIX WITH SCALARWITH SCALAR
For matrix A of order m For matrix A of order m × n × n and scalar number k, the and scalar number k, the matrix of order m matrix of order m × n × n obtained by multiplying each obtained by multiplying each element of A with k is called scalar multiplication of element of A with k is called scalar multiplication of A by k and is denoted by kA.A by k and is denoted by kA.
For A =For A =
2A = 2A =
054
321
__42
642
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MULTIPLICATION OF MATRIX MULTIPLICATION OF MATRIX WITH SCALARWITH SCALAR
For matrix A of order m For matrix A of order m × n × n and scalar number k, the and scalar number k, the matrix of order m matrix of order m × n × n obtained by multiplying each obtained by multiplying each element of A with k is called scalar multiplication of element of A with k is called scalar multiplication of A by k and is denoted by kA.A by k and is denoted by kA.
For A =For A =
2A = 2A =
054
321
__8
642
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MULTIPLICATION OF MATRIX MULTIPLICATION OF MATRIX WITH SCALARWITH SCALAR
For matrix A of order m For matrix A of order m × n × n and scalar number k, the and scalar number k, the matrix of order m matrix of order m × n × n obtained by multiplying each obtained by multiplying each element of A with k is called scalar multiplication of element of A with k is called scalar multiplication of A by k and is denoted by kA.A by k and is denoted by kA.
For A =For A =
2A = 2A =
054
321
_528
642
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MULTIPLICATION OF MATRIX MULTIPLICATION OF MATRIX WITH SCALARWITH SCALAR
For matrix A of order m For matrix A of order m × n × n and scalar number k, the and scalar number k, the matrix of order m matrix of order m × n × n obtained by multiplying each obtained by multiplying each element of A with k is called scalar multiplication of element of A with k is called scalar multiplication of A by k and is denoted by kA.A by k and is denoted by kA.
For A =For A =
2A = 2A =
054
321
_108
642
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MULTIPLICATION OF MATRIX MULTIPLICATION OF MATRIX WITH SCALARWITH SCALAR
For matrix A of order m For matrix A of order m × n × n and scalar number k, the and scalar number k, the matrix of order m matrix of order m × n × n obtained by multiplying each obtained by multiplying each element of A with k is called scalar multiplication of element of A with k is called scalar multiplication of A by k and is denoted by kA.A by k and is denoted by kA.
For A =For A =
2A = 2A =
054
321
02108
642
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MULTIPLICATION OF MATRIX MULTIPLICATION OF MATRIX WITH SCALARWITH SCALAR
For matrix A of order m For matrix A of order m × n × n and scalar number k, the and scalar number k, the matrix of order m matrix of order m × n × n obtained by multiplying each obtained by multiplying each element of A with k is called scalar multiplication of element of A with k is called scalar multiplication of A by k and is denoted by kA.A by k and is denoted by kA.
For A =For A =
2A = 2A =
054
321
0108
642
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MULTIPLICATION OF MATRIX MULTIPLICATION OF MATRIX WITH SCALARWITH SCALAR
For matrix A of order m For matrix A of order m × n × n and scalar number k, the and scalar number k, the matrix of order m matrix of order m × n × n obtained by multiplying each obtained by multiplying each element of A with k is called scalar multiplication of element of A with k is called scalar multiplication of A by k and is denoted by kA.A by k and is denoted by kA.
For A =For A =
-3A = -3A =
054
321
___
__3
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MULTIPLICATION OF MATRIX MULTIPLICATION OF MATRIX WITH SCALARWITH SCALAR
For matrix A of order m For matrix A of order m × n × n and scalar number k, the and scalar number k, the matrix of order m matrix of order m × n × n obtained by multiplying each obtained by multiplying each element of A with k is called scalar multiplication of element of A with k is called scalar multiplication of A by k and is denoted by kA.A by k and is denoted by kA.
For A =For A =
-3A = -3A =
054
321
___
_63
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MULTIPLICATION OF MATRIX MULTIPLICATION OF MATRIX WITH SCALARWITH SCALAR
For matrix A of order m For matrix A of order m × n × n and scalar number k, the and scalar number k, the matrix of order m matrix of order m × n × n obtained by multiplying each obtained by multiplying each element of A with k is called scalar multiplication of element of A with k is called scalar multiplication of A by k and is denoted by kA.A by k and is denoted by kA.
For A =For A =
-3A = -3A =
054
321
___
923
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MULTIPLICATION OF MATRIX MULTIPLICATION OF MATRIX WITH SCALARWITH SCALAR
For matrix A of order m For matrix A of order m × n × n and scalar number k, the and scalar number k, the matrix of order m matrix of order m × n × n obtained by multiplying each obtained by multiplying each element of A with k is called scalar multiplication of element of A with k is called scalar multiplication of A by k and is denoted by kA.A by k and is denoted by kA.
For A =For A =
-3A = -3A =
054
321
__12
323
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MULTIPLICATION OF MATRIX MULTIPLICATION OF MATRIX WITH SCALARWITH SCALAR
For matrix A of order m For matrix A of order m × n × n and scalar number k, the and scalar number k, the matrix of order m matrix of order m × n × n obtained by multiplying each obtained by multiplying each element of A with k is called scalar multiplication of element of A with k is called scalar multiplication of A by k and is denoted by kA.A by k and is denoted by kA.
For A =For A =
-3A = -3A =
054
321
_1512
323
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MULTIPLICATION OF MATRIX MULTIPLICATION OF MATRIX WITH SCALARWITH SCALAR
For matrix A of order m For matrix A of order m × n × n and scalar number k, the and scalar number k, the matrix of order m matrix of order m × n × n obtained by multiplying each obtained by multiplying each element of A with k is called scalar multiplication of element of A with k is called scalar multiplication of A by k and is denoted by kA.A by k and is denoted by kA.
For A =For A =
-3A = -3A =
054
321
01512
323
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PROPERTIES OF SCALAR PROPERTIES OF SCALAR MULTIPLICATIONMULTIPLICATION
k (A + B) = k A + k Bk (A + B) = k A + k B
(-k) A = - (k A) = k (-A)(-k) A = - (k A) = k (-A)
I A = A I = AI A = A I = A
(-1) A = - A(-1) A = - A
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MULTIPLICATION OF MATRICESMULTIPLICATION OF MATRICES
Two matrices can be multiplied only if number Two matrices can be multiplied only if number of columns of first is same as number of rows of columns of first is same as number of rows of the second.of the second.
If A is of order m If A is of order m × n and B is of order n × p, × n and B is of order n × p, then the product AB is a matrix of order m ×then the product AB is a matrix of order m × pp..
m m × × n & nn & n × p × p m × pm × p.. For A = [a For A = [a i ji j] ] m×n m×n and B = [ b and B = [ b j kj k] ] n×p n×p , AB = C , AB = C
with C = [cwith C = [cijij] ] m×p m×p where cwhere ci ki k = = ΣΣ a a ij ij b b jkjk
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MULTIPLICATION OF MATRICES
987
062
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MULTIPLICATION OF MATRICES
987
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MULTIPLICATION OF MATRICES
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MULTIPLICATION OF MATRICES
987
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MULTIPLICATION OF MATRICES
987
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9.90.68.96.67.92.6
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MULTIPLICATION OF MATRICES
987
062
32
96
_8.36.27.32.2
9.90.68.96.67.92.6
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MULTIPLICATION OF MATRICES
987
062
32
96
9.30.28.36.27.32.2
9.90.68.96.67.92.6
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MULTIPLICATION OF MATRICES
8.30.29.36.27.32.2
8.90.69.96.67.92.6
897
062
32
96
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MULTIPLICATION OF MATRICES
8.30.29.36.27.32.2
8.90.69.96.67.92.6
243025
7211775
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TRANPOSE OF MATRIX
For matrix A = [aij] of order m×n, transpose of A is denoted by AT of A/ and 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.
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2331
89
51
A
32385
191
TA
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PROPERTIES OF PROPERTIES OF TRANSPOSE OF MATRICESTRANSPOSE OF MATRICES (A(ATT))TT = A = A (A + B)(A + B)TT= A= ATT + B + BTT
(kA)(kA)TT = k A = k ATT
(AB)(AB)TT = B = BTT A ATT
Every square matrix can be Every square matrix can be expressed as sum of sum of expressed as sum of sum of symmetric and skew-symmetric symmetric and skew-symmetric matrix. A = (A + Amatrix. A = (A + ATT) + (A – A) + (A – ATT))
2
12
1
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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.
273
702
321
073
702
320
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IMPORTANT RESULT ON SYMMETRIC AND SKEW-SYMMETRIC MATRICES
Every square matrix can be expressed as sum of sum of symmetric and skew-symmetric matrix. A =(A + AT) +(A – AT)
All the elements in lead diagonal in skew-symmetric matrix are zero.
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APPLICATION OF MATRICES
Solution of equations in AX=B system using matrix method
(i) If unique solution with
(ii) If and also (adjA)B = 0, Infinite many solutions.
(iii) If (adj A) B 0 No solution.
0A BAX 1,0A
,0A
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Important Problems
1 Construct a 2 3 matrix A with elements given by
2 Find x, y such that
3 If A = diag.(2 -5 9), B = diag.(1 1 -4), find 3A – 2B.
4 Find X and Y if 2X + Y = and X + 2Y =
ji
jiaij
2
5
0
2
0
5
6
1
2
0
2
1
3
6
22
4 yxx
yx
41
23
23
01
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5 Find x if
6 If A = show that
.
7 Show that
8 If and Find K so that .
9 Show that is symmetric or skew-symmetric according as A is symmetric or skew-symmetric.
02
1
2315
152
231
11
x
x
21
130752 AA
0
1
1
1
1
1
1
1
22
2
2
2
2
2
w
w
ww
ww
ww
ww
ww
ww
21
13A KIAA 52
ABB
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10 Express A = as sum of symmetric
and skew-symmetric matrices.
11 If A = find
12 Find X if
13 Solve using matrix method
x + 2y + z = 7, x + 3 z = 11, 2 x – 3 y = 1.
542
354
323
23
64&
52
23 1B 1AB
41
12
12
11
57
23X
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Address of the subject related websites
http://www.netsoc.tcd.ie/~jgilbert/maths_site/applets/algebra/matrix_inversion.html
http://www.ping.be/~ping1339/matr.htm
http://mathworld.wolfram.com/Matrix.html
http://en.wikipedia.org/wiki/Matrices
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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.