Question

Given is the following coefficient matrix of a homogeneous system of linear equations over R (the set of real numbers): $\begin{pmatrix} -1 & -3 & -7 & 0 & 9 \ -2 & -12 & -6 & 7 & 10 \ -4 & -30 & -4 & 15 & 22 \ 4 & 30 & 4 & -15 & -20 \ -3 & -15 & -13 & 7 & 21 \end{pmatrix}$ In the lecture, we defined the following row transformations: $\tau_{ij}$: Swap rows $i$ and $j$ $\alpha_{ij}(c)$ for $i \neq j$: Add $c$ times row $j$ to row $i$ $\mu_i(c)$ for $c \neq 0$: Multiply row $i$ by $c$ Apply the following operations sequentially to the matrix: $\tau_{15}$, $\alpha_{12}(1)$, $\alpha_{13}(1)$, $\alpha_{23}(-2)$, $\mu_3(-1/4)$. Let A = ($a_{ij}$)1$\leq i, j \leq 5$ be the resulting matrix. Furthermore, let Z be a matrix in row echelon form, obtained from A through elementary row transformations. 1) The number of levels (or steps) r of Z is: 2) The number of elements in the solution set of the given system of equations is: (Enter either a number or the letter \"u\" if the set is infinite.) 3) How many rows of zeros does the normal form of the matrix contain? 4) What is the value of a14? 5) The number of free variables in Z is:

          Given is the following coefficient matrix of a homogeneous system of linear equations over
R (the set of real numbers):
$\begin{pmatrix} -1 & -3 & -7 & 0 & 9 \ -2 & -12 & -6 & 7 & 10 \ -4 & -30 & -4 & 15 & 22 \ 4 & 30 & 4 & -15 & -20 \ -3 & -15 & -13 & 7 & 21 \end{pmatrix}$
In the lecture, we defined the following row transformations:
$\tau_{ij}$: Swap rows $i$ and $j$
$\alpha_{ij}(c)$ for $i \neq j$: Add $c$ times row $j$ to row $i$
$\mu_i(c)$ for $c \neq 0$: Multiply row $i$ by $c$
Apply the following operations sequentially to the matrix:
$\tau_{15}$, $\alpha_{12}(1)$, $\alpha_{13}(1)$, $\alpha_{23}(-2)$, $\mu_3(-1/4)$.
Let A = ($a_{ij}$)1$\leq i, j \leq 5$ be the resulting matrix.
Furthermore, let Z be a matrix in row echelon form, obtained from A through elementary row
transformations.
1) The number of levels (or steps) r of Z is:
2) The number of elements in the solution set of the given system of
equations is:
(Enter either a number or the letter \"u\" if the set is infinite.)
3) How many rows of zeros does the normal form of the matrix contain?
4) What is the value of a14?
5) The number of free variables in Z is:

Given is the following coefficient matrix of a homogeneous system of linear equations over
R (the set of real numbers):
< p m a t r i x >
In the lecture, we defined the following row transformations:
τij: Swap rows i and j
αij(c) for i ≠ j: Add c times row j to row i
(c) for c ≠ 0: Multiply row i by c
Apply the following operations sequentially to the matrix:
τ15, α12(1), α13(1), α23(-2), μ3(-1/4).
Let A = (aij)1≤ i, j ≤ 5 be the resulting matrix.
Furthermore, let Z be a matrix in row echelon form, obtained from A through elementary row
transformations.
1) The number of levels (or steps) r of Z is:
2) The number of elements in the solution set of the given system of
equations is:
(Enter either a number or the letter üïf the set is infinite.)
3) How many rows of zeros does the normal form of the matrix contain?
4) What is the value of a14?
5) The number of free variables in Z is:

Added by Shannon J.

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