Question

Compute the determinant of the following matrix by first putting the matrix in reduced echelon form:

[2 1 -1 0 3

1 0 2 4 -2

4 -2 0 5 1

3 -3 2 1 7

-2 1 0 -3 5]

Answer #1

Find the reduced row echelon form of the following matrices.
Interpret your result by giving the solutions of the systems whose
augmented matrix is the one given.
[ 0 4 7 0
2 1 0 0
0 3 1 -4 ]

Argue that the only way for a square matrix Ain reduced echelon
form Arr to have a non-zero determinant is if Arr=I, the identity
matrix.

Solve and explain the process of solving the following system
with matrix reduced row-echelon form. Last, explain your results
for the following problem:
3x – 4y + 4z =7
x – y – 2z = 2
2x – 3y + 6z = 5

Write the system of equations as an augmented matrix. Then solve
the system by putting the matrix in reduced row echelon form.
x+2y−z=-10
2x−3y+2z=2
x+y+3z=0

If the reduced row echelon form of an m*n matrix A has a pivot
in every row, explain why the columns of A must span R^m

T12. Suppose that A is a square matrix. Using the definition of
reduced row-echelon form (Definition RREF) carefully, give a proof
of the following equivalence: Every column of A is a pivot column
if and only if A is the identity matrix (Definition IM).
http://linear.ups.edu/html/section-NM.html

Compute the determinant of the following matrix using row
reduction:
| 14 -6 0 -1 |
| -37 17 -2 5 |
| 25 -11 1 -4 |
| -5 2 0 1 |

Compute the determinant using a cofactor expansion across the
first row. Also compute the determinant by a cofactor expansion
down the second column.
|2 0 3
|2 4 3
|0 5 -1

Solve the following systems by forming the augmented matrix and
reducing to reduced row echelon form. In each case decide whether
the system has a unique solution, infinitely many solutions or no
solution. Show pivots in squares. Describe the solution set.
-3x1+x2-x3=10
x2+4X3=12
-3x1+2x2+3x3=11

3. Write the matrix in row-echelon form:
1
2
-1
3
3
7
-5
14
-2
-1
-3
8

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