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![TYPES OF MATRICES
Negative Matrix
A negative matrix of A =[aij] denoted by –A
where -A =[-aij]
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201
164
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201
AA](https://image.slidesharecdn.com/lavc-170714110720/75/Matrices-and-System-of-Linear-Equations-ppt-10-2048.jpg)
![TYPES OF MATRICES
Transpose of Matrix
If A =[aij] is an m x n matrix, then the
transpose of A, AT =[aij]T is the n x m matrix
defined by [aij] = [aji]T
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411
321
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T
AA](https://image.slidesharecdn.com/lavc-170714110720/75/Matrices-and-System-of-Linear-Equations-ppt-13-2048.jpg)
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Matrices and System of Linear Equations ppt
- 1. Matrices and System of LinearEquations
- 2. DHRUV MISTRY160280106055 DHRUVIL CHAVDA 160280106015 SAVAN BARIA 160280106005 DHARMIK DAVE 160280106022 YASH JANI 160280106037 ATRI BHATT 160280106007 CHIRAG 160280106051 VEDANT 160280106047 YASH JAIN 160280106034 AKHIL 160280106059 SANDEEP 160280106046
- 3. INTRODUCTION • Definition A matrixis a rectangular array of elements or entries aij involving m rows and n columns nmij ijiii j j j a aaaa aaaa aaaa aaaa A 321 3333231 2232221 1131211 Rows, m Columns, n
- 4. INTRODUCTION • Definition i. 2matrices and are said to be equal iff m = r and n = s then A = B. ii. If aij for i = j, then the entries a11,a22,a33,… are called the diagonal of matrix A nmij MaA nmij MbB
- 5. TYPES OF MATRICES SquareMatrix Matrix with order n x n 33 987 654 321 43 21 BA nm
- 6. TYPES OF MATRICES DiagonalMatrix Matrix with order n x n with aij ≠ 0 and aij = 0 for i ≠ j 200 010 001 A
- 7. TYPES OF MATRICES ScalarMatrix A diagonal matrix in which the diagonal elements are equal, aii = k and aij = 0 for i ≠ j where k is a scalar 2, 100 010 001 2 200 020 002 kA
- 8. TYPES OF MATRICES IdentityMatrix A diagonal matrix in which the diagonal elements are ‘1’, aii = 1 and aij ≠ 0 for i ≠ j 10 01 100 010 001 BA
- 9. TYPES OF MATRICES ZeroMatrix A matrix which contains only zero elements, aij = 0 00 00 000 000 000 BA
- 10. TYPES OF MATRICES NegativeMatrix A negative matrix of A =[aij] denoted by –A where -A =[-aij] 164 300 201 164 300 201 AA
- 11. TYPES OF MATRICES UpperTriangular Matrix If every elements below the diagonal is zero or aij = 0, i > j 100 310 221 A DIAGONAL
- 12. TYPES OF MATRICES LowerTriangular Matrix If every elements above the diagonal is zero or aij = 0, i < j 143 012 001 A DIAGONAL
- 13. TYPES OF MATRICES Transposeof Matrix If A =[aij] is an m x n matrix, then the transpose of A, AT =[aij]T is the n x m matrix defined by [aij] = [aji]T 152 411 321 143 512 211 T AA
- 14. TYPES OF MATRICES PropertiesTransposition Operation • Let A and B matrices and k, . Then, TTT TT TT BABAiii kAkAii AAi ) ) ) Rk
- 15. TYPES OF MATRICES Example1: • If and , find T T T ABiii Bii Ai ) 2) ) 254 321 A 2 3 1 B
- 16. TYPES OF MATRICES Answer1: 231) 4622) 2 5 4 3 2 1 ) T T T ABiii Bii Ai
- 17. TYPES OF MATRICES SymmetricMatrix If AT = A, where the elements obey the rule aij = aji 324 221 415 324 221 415 T AA
- 18. TYPES OF MATRICES SkewSymmetric Matrix If AT = - A, where the elements obey the rule aij = - aji, so that the diagonal must contain zeroes. 074 701 410 A
- 19. TYPES OF MATRICES SkewSymmetric Matrix AA A T 074 701 410 074 701 410 074 701 410
- 20. TYPES OF MATRICES RowEchelon Form (REF) Matrix A is said to be in REF if it satisfies the following properties: • Rows consisting entirely zeroes occur at the bottom of the matrix. • For each row that doesn’t consist entirely of zeroes, the 1st nonzero is 1. • For each non zero row, number 1 appear to the right of the leading 1 of the previous row.
- 21. TYPES OF MATRICES 100 310 421 000 710 411 B A LEADING 1 ZERO ROW AT THE BOTTOM LEADING 1
- 22. TYPES OF MATRICES ReducedRow Echelon Form (RREF) Matrix A is said to be in REF if it satisfies the following properties: • Rows consisting entirely zeroes occur at the bottom of the matrix. • For each row that doesn’t consist entirely of zeroes, the 1st nonzero is 1. • For each non zero row, number 1 appear to the right of the leading 1 of the previous row. • If a column contains a leading 1, then all other entries in the column are zero
- 23. TYPES OF MATRICES 100 010 001 000 710 401 B A LEADING 1 ZERO ROW AT THE BOTTOM LEADING 1
- 24. Types of Solutions ConsistentSystem One solution Consistent System Infinite solutions Inconsistent System No solution
- 25. A linear equationis an equation that can be written in the form: The coefficients ai and the constant b can be real or complex numbers. A Linear System is a collection of one or more linear equations in the same variables. Here are a few examples of linear systems: bxaxaxa nn2211 4z2y2x 2zy3x2 1zyx 4xx4x 2xxx2x 1xx2x 532 5432 421 1zy2x 7z7y4x3 Any system of linear equations can be put into matrix form: bxA The matrix A contains the coefficients of the variables, and the vector x has the variables as its components. For example, for the first system above the matrix version would be: bxA 4 2 1 z y x 221 132 111
- 26. Prepared by VinceZaccone For Campus Learning Assistance Services at UCSB Here is the next system. The basic pattern is to start at the upper left corner, then use row operations to get zeroes below, then work counterclockwise until the matrix is in REF. 3 0 1 310 310 111 RRR R2RR 4 2 1 221 132 111 4z2y2x 2zy3x2 1zyx 13 * 3 12 * 2 3 0 1 000 310 111 RRR3 0 1 310 310 111 23 * 3 At this point you might notice a problem. That last row doesn’t make sense. It might help to write out the equation that the last row represents. It says 0x+0y+0z=3. Are there any values of x, y and z that make this equation work? (the answer is NO!) This system is called INCONSISTENT because we arrive at a contradiction during the solution procedure. This means that the system has no solution.
- 27. This line isthe intersection of a pair of the planes This line is the intersection of a different pair of the planes Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB No Solution If the system is inconsistent there will be no solutions. In this case there will be a contradiction that appears during the solution process. This is the reduced matrix (actually we could go one step further and get a zero up in row 1). Notice that we got a row of zeroes in the left part of the augmented matrix. When this happens the system will either be inconsistent, like this one, or we will have a free variable (infinite # of solutions). 4z2y2x 2zy3x2 1zyx 3 0 1 000 310 111
- 28. Consistent or InconsistentSystem? 2100 5010 1001 𝑦 + 3𝑧 = 2 𝑧 = 𝑎𝑛𝑦 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 2000 7010 4001 0000 2310 2501 𝑥 + 5𝑧 = 2 (𝑥 = 2 − 5𝑧, 𝑦 = 2 − 3𝑧, 𝑧 = 𝑎𝑛𝑦 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟) 𝑦 = 5 𝑧 = 2 𝑥 = 1 𝑦 = 7 0 = 2 𝑥 = 4 𝑁𝑜 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛: 𝐼𝑛𝑐𝑜𝑛𝑠𝑖𝑠𝑡𝑒𝑛𝑡 𝑠𝑦𝑠𝑡𝑒𝑚 𝐼𝑛𝑓𝑖𝑛𝑖𝑡𝑒 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛𝑠: 𝐶𝑜𝑛𝑠𝑖𝑠𝑡𝑒𝑛𝑡 𝑠𝑦𝑠𝑡𝑒𝑚 {(𝑥, 𝑦, 𝑧)|𝑥 = 2 − 5𝑧, 𝑦 = 2 − 3𝑧, 𝑧 = 𝑎𝑛𝑦 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟} 𝑜𝑟
- 29. Inverse Matrix Method Givena set of linear equations x + 2y = 4 3x − 5y = 1 It can be written in matrix form as… = AX = B This is the matrix form of the linear equations 1 2 3 −5( ) x y( ) 4 1( )
- 30. Given that… AX =B A-1AX = A-1B (A−1A = I, the identity matrix and multiplying any matrix by I leaves the matrix unchanged) X = A−1B
- 31.