6.2.2) the given matrix A can easily be manipulated into an upper triangular matrix. Once you have the new matrix B (the upper triangular matrix), the determinant of A is easily obtained from the formulas given in the text.

6.2.8) The given matrix A can be easily manipulated by a number of row swapping. If you swap the rows and make sure the given first row is the last row, you can easily reduce the last column of the new matrix by again manipulating the last row (original first row). After that you should have the rref.

6.3.24) To find the values of x, y, and z you must use Cramer's rule. For solving for x, take a 3x3 (matrix from the given values) of the values associated with y and z, and the last column of the matrix. Find its determinant, then divide by the determinant of the 3x3 matrix of values of all three variables. To find y and z, use the same method except leave out the column containing the variable that you are looking for.

2x+3y =8...........(3 0 8)......(2 3 0)

4y+5z=3 ==> x= det(4 5 3)../det(0 4 5)

6x+ +7z=-1..........(0 7 -1).....(6 0 7)

• the numbers in parenthesis are the numbers in the matrix*

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Basic linear algebra uncovers and clarifies very important geometry and algebra. Dr. Paul Garrett