# Equations and Matrices

17. Solve the system of equations by the Gaussian elimination method.

x- 3y + z= 8

2x- 5y -3 z= 2

x + 4y + z= 1

18. Find the inverse of the given matrix.

1 2

-2 -3

19. Evaluate the determinant by expanding by cofactors.

-2 3 2

1 2 -3

-4 -2 1

20. Solve the system of equations by using Cramer's Rule.

2x + 5 y= 9

5x + 7y =8

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## SOLUTION This solution is **FREE** courtesy of BrainMass!

17. Solve the system of equations by the Gaussian elimination method.

x- 3y + z= 8 -------------1

2x- 5y -3 z= 2 -----------2

x + 4y + z= 1 ------------3

Eliminate x from equation 2 and 3

Equation 2 - 2 Equation 1 is

(2x- 5y -3 z= 2 ) - 2 (x- 3y + z= 8)

or y-5z= -14 -----new equation 2

Equation 3 -Equation 1 is

(x + 4y + z= 1) -(x- 3y + z= 8)

or 7y = -7

0r y= -1 new equation 3

Substituting this value of y in new equation 2

y-5z= -14

-1-5z=-14

0r -5z=-13

0r z=-13/-5=2.6

y=-1

z=2.6

Substituting these values in Equation 1

x- 3y + z= 8

x-3(-1)+2.6=8

x=8-3-2.6=2.4

x=2.4

y=-1

z=2.6

Answer

18. Find the inverse of the given matrix.

elements of Inverse of a matrix = (cofactor of ajk in determinant of A) / det [A]

Determinant of matrix= (1* -3)-(2*-2)=-3+4=1

Cofactor of 1 is -3

Cofactor of 2 is -(-2)=2

Cofactor of -2 is -(2)=-2

Cofactor of -3 is 1

Inverse = 1/ det [A] * |-3 2|

|-2 1 |

=1/1 * |-3 2 |

| -2 1|

Inverse of the matrix

=|-3 2 |

| -2 1|

19. Evaluate the determinant by expanding by cofactors.

Determinant of matrix=

+ (-2) * {(2* 1)-(-3*-2) } - (3) * {(1* 1)-(-3*-4) }+ (2) * {(1* -2)-(2 *-4) }

=+ (-2) * {(2-6)} - (3) * {(1-12}+ (2) * {(-2+8) }

=8 +33+12= 53

Answer: 53

20. Solve the system of equations by using Cramer's Rule.

The system of equation is solved using the Cramer's rule

det is the determinant

D= det | 2 5 |

| 5 7 |

=2*7-5*5=-11

D1= det | 9 5 |

| 8 7 |

=9*7-5*8= 23

D2= det | 2 9 |

| 5 8 |

=2*8-5*9= -29

x=D1/D= 23/(-11)= - 23/11

y= D2/D= -29/(-11)= 29 /11

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