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Write down matrix A^-1 as the product of elementary matrices if: A=01211021-1
Related Topics
Wize University Linear Algebra Textbook > Matrices
Elementary Matrices
5 Activities
Write down matrix
A
−
1
A^{-1}
A
−
1
as the product of elementary matrices if:
A
=
[
0
1
2
1
1
0
2
1
−
1
]
A=\begin{bmatrix}0&1&2\\1&1&0\\2&1&-1\end{bmatrix}
A
=
0
1
2
1
1
1
2
0
−
1
E
1
=
[
0
1
0
1
0
0
0
0
1
]
E
2
=
[
1
0
0
0
1
0
−
2
0
1
]
E
3
=
[
1
−
1
0
0
1
0
0
0
1
]
E
4
=
[
1
0
0
0
1
0
−
2
0
1
]
E
5
=
[
1
0
0
0
1
2
0
0
1
]
E
6
=
[
1
0
2
0
1
0
0
0
1
]
\begin{array}{ll} E_1=\begin{bmatrix}0&1&0\\1&0&0\\0&0&1\end{bmatrix} & E_2=\begin{bmatrix}1&0&0\\0&1&0\\-2&0&1\end{bmatrix}\\ E_3=\begin{bmatrix}1&-1&0\\0&1&0\\0&0&1\end{bmatrix} & E_4=\begin{bmatrix}1&0&0\\0&1&0\\-2&0&1\end{bmatrix}\\ E_5=\begin{bmatrix}1&0&0\\0&1&2\\0&0&1\end{bmatrix} & E_6=\begin{bmatrix}1&0&2\\0&1&0\\0&0&1\end{bmatrix} \end{array}
E
1
=
0
1
0
1
0
0
0
0
1
E
3
=
1
0
0
−
1
1
0
0
0
1
E
5
=
1
0
0
0
1
0
0
2
1
E
2
=
1
0
−
2
0
1
0
0
0
1
E
4
=
1
0
−
2
0
1
0
0
0
1
E
6
=
1
0
0
0
1
0
2
0
1
E
1
=
[
1
0
0
0
1
0
−
2
0
1
]
E
2
=
[
0
1
0
1
0
0
0
0
1
]
E
3
=
[
1
−
1
0
0
1
0
0
0
1
]
E
4
=
[
1
0
0
0
1
0
−
2
0
1
]
E
5
=
[
1
0
0
0
1
−
2
0
0
1
]
E
6
=
[
1
0
2
0
1
0
0
0
1
]
\begin{array}{ll} E_1=\begin{bmatrix}1&0&0\\0&1&0\\-2&0&1\end{bmatrix} & E_2=\begin{bmatrix}0&1&0\\1&0&0\\0&0&1\end{bmatrix}\\ E_3=\begin{bmatrix}1&-1&0\\0&1&0\\0&0&1\end{bmatrix} & E_4=\begin{bmatrix}1&0&0\\0&1&0\\-2&0&1\end{bmatrix}\\ E_5=\begin{bmatrix}1&0&0\\0&1&-2\\0&0&1\end{bmatrix} & E_6=\begin{bmatrix}1&0&2\\0&1&0\\0&0&1\end{bmatrix} \end{array}
E
1
=
1
0
−
2
0
1
0
0
0
1
E
3
=
1
0
0
−
1
1
0
0
0
1
E
5
=
1
0
0
0
1
0
0
−
2
1
E
2
=
0
1
0
1
0
0
0
0
1
E
4
=
1
0
−
2
0
1
0
0
0
1
E
6
=
1
0
0
0
1
0
2
0
1
E
1
=
[
0
1
0
1
0
0
0
0
1
]
E
2
=
[
1
0
0
0
1
0
−
2
0
1
]
E
3
=
[
1
−
1
0
0
1
0
0
0
1
]
E
4
=
[
1
0
0
0
1
0
0
1
1
]
E
5
=
[
1
0
0
0
1
−
2
0
0
1
]
E
6
=
[
1
0
2
0
1
0
0
0
1
]
\begin{array}{ll} E_1=\begin{bmatrix}0&1&0\\1&0&0\\0&0&1\end{bmatrix} & E_2=\begin{bmatrix}1&0&0\\0&1&0\\-2&0&1\end{bmatrix}\\ E_3=\begin{bmatrix}1&-1&0\\0&1&0\\0&0&1\end{bmatrix} & E_4=\begin{bmatrix}1&0&0\\0&1&0\\0&1&1\end{bmatrix}\\ E_5=\begin{bmatrix}1&0&0\\0&1&-2\\0&0&1\end{bmatrix} & E_6=\begin{bmatrix}1&0&2\\0&1&0\\0&0&1\end{bmatrix} \end{array}
E
1
=
0
1
0
1
0
0
0
0
1
E
3
=
1
0
0
−
1
1
0
0
0
1
E
5
=
1
0
0
0
1
0
0
−
2
1
E
2
=
1
0
−
2
0
1
0
0
0
1
E
4
=
1
0
0
0
1
1
0
0
1
E
6
=
1
0
0
0
1
0
2
0
1
E
1
=
[
0
1
0
1
0
0
0
0
1
]
E
2
=
[
1
0
0
0
1
0
−
2
0
1
]
E
3
=
[
1
0
2
0
1
0
0
0
1
]
E
4
=
[
1
0
0
0
1
0
−
2
0
1
]
E
5
=
[
1
0
0
0
1
−
2
0
0
1
]
E
6
=
[
1
−
1
0
0
1
0
0
0
1
]
\begin{array}{ll} E_1=\begin{bmatrix}0&1&0\\1&0&0\\0&0&1\end{bmatrix} & E_2=\begin{bmatrix}1&0&0\\0&1&0\\-2&0&1\end{bmatrix}\\ E_3=\begin{bmatrix}1&0&2\\0&1&0\\0&0&1\end{bmatrix} & E_4=\begin{bmatrix}1&0&0\\0&1&0\\-2&0&1\end{bmatrix}\\ E_5=\begin{bmatrix}1&0&0\\0&1&-2\\0&0&1\end{bmatrix} & E_6=\begin{bmatrix}1&-1&0\\0&1&0\\0&0&1\end{bmatrix} \end{array}
E
1
=
0
1
0
1
0
0
0
0
1
E
3
=
1
0
0
0
1
0
2
0
1
E
5
=
1
0
0
0
1
0
0
−
2
1
E
2
=
1
0
−
2
0
1
0
0
0
1
E
4
=
1
0
−
2
0
1
0
0
0
1
E
6
=
1
0
0
−
1
1
0
0
0
1
None of the Above.
I don't know
Check Submission
More Elementary Matrices Questions:
Write down matrix
A
−
1
A^{-1}
A
−
1
as the product of elementary matrices if:
A
=
[
0
1
2
1
1
0
2
1
−
1
]
A=\begin{bmatrix}0&1&2\\1&1&0\\2&1&-1\end{bmatrix}
A
=
0
1
2
1
1
1
2
0
−
1
Elementary matrices
Which of the following are elementary matrices?
A
=
[
1
0
0
0
1
0
0
0
−
3
]
B
=
[
0
0
0
1
0
0
1
0
0
1
0
0
1
0
0
0
]
A = \begin{bmatrix} 1 & 0 & 0\\0 & 1 & 0\\0 & 0 & -3\end{bmatrix} \quad B = \begin{bmatrix} 0 & 0 & 0 & 1\\0 & 0 & 1 & 0\\0 & 1 & 0 & 0\\1 & 0 & 0 & 0\end{bmatrix}
A
=
1
0
0
0
1
0
0
0
−
3
B
=
0
0
0
1
0
0
1
0
0
1
0
0
1
0
0
0
C
=
[
0
0
1
0
1
0
1
0
0
]
D
=
[
1
0
−
7
1
]
C = \begin{bmatrix} 0 & 0 & 1\\0 & 1 & 0\\1 & 0 & 0\end{bmatrix} \quad D = \begin{bmatrix} 1 & 0 \\-7 & 1 \end{bmatrix}
C
=
0
0
1
0
1
0
1
0
0
D
=
[
1
−
7
0
1
]
Example: Elementary Matrix
Find an elementary matrix
E
E
E
such that
A
=
E
B
A=EB
A
=
E
B
if:
A
=
[
2
4
6
−
1
−
1
−
1
1
−
1
3
]
A=\left[\begin{array}{rrr} 2&4&6\\ -1&-1&-1\\ 1&-1&3 \end{array}\right]
A
=
2
−
1
1
4
−
1
−
1
6
−
1
3
and
B
=
[
2
4
6
−
1
−
1
−
1
3
1
5
]
B=\left[\begin{array}{rrr} 2&4&6\\ -1&-1&-1\\ 3&1&5 \end{array}\right]
B
=
2
−
1
3
4
−
1
1
6
−
1
5
Which of the following are elementary matrices?
A
=
[
1
0
0
0
1
0
0
0
−
3
]
B
=
[
0
0
0
1
0
0
1
0
0
1
0
0
1
0
0
0
]
A = \begin{bmatrix} 1 & 0 & 0\\0 & 1 & 0\\0 & 0 & -3\end{bmatrix} \quad B = \begin{bmatrix} 0 & 0 & 0 & 1\\0 & 0 & 1 & 0\\0 & 1 & 0 & 0\\1 & 0 & 0 & 0\end{bmatrix}
A
=
1
0
0
0
1
0
0
0
−
3
B
=
0
0
0
1
0
0
1
0
0
1
0
0
1
0
0
0
C
=
[
0
0
1
0
1
0
1
0
0
]
D
=
[
1
0
−
7
1
]
C = \begin{bmatrix} 0 & 0 & 1\\0 & 1 & 0\\1 & 0 & 0\end{bmatrix} \quad D = \begin{bmatrix} 1 & 0 \\-7 & 1 \end{bmatrix}
C
=
0
0
1
0
1
0
1
0
0
D
=
[
1
−
7
0
1
]
Practice Question: Matrix of an ERO
Let
E
E
E
be the elementary matrix obtained from
I
2
I_2
I
2
by multiplying
R2
by 1/3. Let
B
B
B
be any
2
×
2
2\times2
2
×
2
matrix. Show that
E
B
EB
E
B
is the result of multiplying
R2
of
B
B
B
by 1/3. Find the inverse of
E
E
E
and show that it is an elementary matrix.
Matrix Multiplication
Express the following matrix
A
‾
\bcb{ \A }
A
as a product of five (or fewer) elementary matrices); repeat for
A
‾
−
1
\bcb{ \minv{\A} }
A
−
1
.
A
‾
=
[
−
3
0
0
2
0
1
0
1
0
]
\bcb{ \A=\begin{bmatrix} -3&0&0\\2&0&1\\0&1&0 \end{bmatrix} }
A
=
−
3
2
0
0
0
1
0
1
0
Practice: Elementary Matrix
Find an elementary matrix
E
E
E
such that
E
B
=
C
EB=C
E
B
=
C
if:
B
=
[
1
1
3
0
0
3
6
4
]
B=\left[\begin{array}{rr} 1&1\\ 3&0\\ 0&3\\ 6&4 \end{array}\right]
B
=
1
3
0
6
1
0
3
4
and
C
=
[
10
1
3
0
0
3
6
4
]
C=\left[\begin{array}{rr} 10&1\\ 3&0\\ 0&3\\ 6&4 \end{array}\right]
C
=
10
3
0
6
1
0
3
4
Matrix Multiplication
e.g. Express the following matrix
A
‾
\underline{A}
A
as a product of five (or fewer) elementary matrices); repeat for
A
‾
−
1
\!\underline{A}^{−1}\!
A
−
1
,