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picture1_Matrices And Determinants Pdf 174113 | Ela7 Item Download 2023-01-27 20-00-07


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File: Matrices And Determinants Pdf 174113 | Ela7 Item Download 2023-01-27 20-00-07
elementarylinearalgebra set7 determinants systems of linear equations 1 write the laplace expansions of the given determinants along indicated rows or columns do not perform calculations of the determinants 1 2 ...

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                              ELEMENTARYLINEARALGEBRA–SET7
                                    Determinants, systems of linear equations
                 1. Write the Laplace expansions of the given determinants along indicated rows or
                    columns (do not perform calculations of the determinants)
                                                                                    
                                                 1  2 3 0          1 3    2    1 
                                 −1 4     3                                       
                                                 0 1 0 2           2 4 −1      0 
                                 −3 1     0 ,               ,                     
                                                 2  3 3 0        −1 0     2    0 
                                  2 5 −2                                          
                 2. Calculate the determinants     1  2 3 1          3 2    5 −1 
                                                                                  
                                                                   1 0     2 1 
                                                 1 2     3                    
                                    −2 4                         2 1 −1 3 
                                           ,   −1 1 −1 ,                      
                                    −3 1                        −1 0     2 0 
                                                   2 1     3                    
                                                                     3 2     1 1 
                 3. Using the properties of the determinants, justify that the following matrices are
                    singular                                              
                                                              1 3 2 1 
                                             1    2    3                
                                                              4 2 1 3 
                                             0    1 −1 ,                
                                                              3 3 1 2 
                                            −2 −4 −6                    
                                                                0 4 2 0 
                 4. Compute the determinants in Problem 2, using the Gauss algorithm.
                 5. Using the cofactor formula, compute the inverses of the following matrices:
                                                1 0      0      0 1 0 0 
                                     −2 4                           2 0 0 0 
                                              ,   3 1      0 ,                  
                                     −3 1            2 2 −1         0 0 0 3 
                                                                       0 0 4 0
                 6. Using inverse matrices, solve the following matrix equations:
                             3 5           0 1      1          1 0       0         1
                       (a)     1 2    · X =    2 3 −1      ,  (b)  3 1       0 ·X = 3 
                                                                      2 2 −1                2
                 7. Applying Cramer’s Rule to the following systems of equations, compute the indi-
                    cated unknown:
                                                             x+y+2z=−1
                       (a)    2x−y=0 , unknowny (b)  2x−y+2z =−4 , unknownx
                              3x+2y=5                         4x+y+4z=−2
                                                        1
                   8. Applying the Gauss elimination method, solve the following systems of equations
                                                   x + 2y + z = 3
                                                  3x + 2y + z = 3
                                                x − 2y − 5z = 1
                                               x + 2y + 4z − 3t = 0
                                              3x + 5y + 6z − 4t = 1
                                                 4x + 5y − 2z + 3t = 1
                   9. Applying the Kronecker-Capelli theorem, show that the system
                                              x + 2y + 3z − t = −1
                                             3x + 6y + 7z + t =                     5
                                             2x + 4y + 7z − 4t = −6
                      has infinitely many solutions and then solve this system.
                  10. Applying the Kronecker-Capelli theorem, show that the system
                                              x − y − 2z + 2t = −2
                                             
                                             
                                             5x − 3y − z + t =                      3
                                             2x + y − z + t = 1
                                             
                      is inconsistent.       3x − 2y + 2z − 2t = −4
                Romuald Lenczewski
                                                              2
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...Elementarylinearalgebra set determinants systems of linear equations write the laplace expansions given along indicated rows or columns do not perform calculations calculate using properties justify that following matrices are singular compute in problem gauss algorithm cofactor formula inverses inverse solve matrix a x b applying cramer s rule to indi cated unknown y z unknowny unknownx elimination method t kronecker capelli theorem show system has innitely many solutions and then this is inconsistent romuald lenczewski...

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