Question

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.

Answer #1

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

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

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

Show that the nonzero rows of a reduced row echelon form A form
a basis of the row space R (A). Hint: Name the positions of pivotal
entries by indices of the form (i, ki) with ki+1 > ki .

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 ]

2X1-X2+X3+7X4=0
-1X1-2X2-3X3-11X4=0
-1X1+4X2+3X3+7X4=0
a. Find the reduced row - echelon form of the coefficient
matrix
b. State the solutions for variables X1,X2,X3,X4 (including
parameters s and t)
c. Find two solution vectors u and v such that the solution
space is \
a set of all linear combinations of the form su + tv.

Answer all of the questions true or false:
1.
a) If one row in an echelon form for an augmented matrix is [0 0 5
0 0]
b) A vector b is a linear combination of the columns of a matrix A
if and only if the
equation Ax=b has at least one solution.
c) The solution set of b is the set of all vectors of the form u =
+ p + vh
where vh is any solution...

1. Let a,b,c,d be row vectors and form the matrix A whose rows
are a,b,c,d. If by a sequence of row operations applied to A we
reach a matrix whose last row is 0 (all entries are 0) then:
a. a,b,c,d are linearly dependent
b. one of a,b,c,d must be 0.
c. {a,b,c,d} is linearly independent.
d. {a,b,c,d} is a basis.
2. Suppose a, b, c, d are vectors in R4 . Then they form a...

Prove that for a square n ×n matrix A, Ax = b (1) has one and
only one solution if and only if A is invertible; i.e., that there
exists a matrix n ×n matrix B such that AB = I = B A.
NOTE 01: The statement or theorem is of the form P iff Q, where
P is the statement “Equation (1) has a unique solution” and Q is
the statement “The matrix A is invertible”. This means...

Let A be square matrix prove that A^2 = I if and only if
rank(I+A)+rank(I-A)=n

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