- #1
nietzsche
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I had a quiz last Thursday about matrices, and I thought I knew what I was doing, but when I got there, nothing made sense to me. I'm pretty frustrated because I'm having a lot of trouble grasping any of the concepts I'm learning in linear algebra, while I'm doing fine in my "regular" math courses. Linear algebra just seems so abstract and...wrong...
Write A as a product of elementary matrices. (Hint: First, write the inverse as a product of elementary matrices.)
[tex]
A =
\left[
\begin{array}{cc}
1 & 2 \\
3 & 4
\end{array}
\right]
[/tex]
I just don't understand why any of this works. I know how to find the inverse using Gauss-Jordan elimination. I wrote the partitioned matrix with A on the left and the 2x2 identity on the right. Then I performed the following row operations:
R2 - 3R1
R1 + R2
(-1/2)R2
and I got the following partitioned matrix:
[tex]
\left[
\begin{array}{cc|cc}
1 & 0 & -2 & 1 \\
0 & 1 & \frac{3}{2} & -\frac{1}{2}
\end{array}
\right]
[/tex]
I simply don't understand the concept of how a matrix can be a product of elementary matrices. I don't see how a matrix can correspond to an elementary row operation. Please help me understand! I'm seriously beginning to go a bit loopy. Thanks.
Homework Statement
Write A as a product of elementary matrices. (Hint: First, write the inverse as a product of elementary matrices.)
[tex]
A =
\left[
\begin{array}{cc}
1 & 2 \\
3 & 4
\end{array}
\right]
[/tex]
Homework Equations
The Attempt at a Solution
I just don't understand why any of this works. I know how to find the inverse using Gauss-Jordan elimination. I wrote the partitioned matrix with A on the left and the 2x2 identity on the right. Then I performed the following row operations:
R2 - 3R1
R1 + R2
(-1/2)R2
and I got the following partitioned matrix:
[tex]
\left[
\begin{array}{cc|cc}
1 & 0 & -2 & 1 \\
0 & 1 & \frac{3}{2} & -\frac{1}{2}
\end{array}
\right]
[/tex]
I simply don't understand the concept of how a matrix can be a product of elementary matrices. I don't see how a matrix can correspond to an elementary row operation. Please help me understand! I'm seriously beginning to go a bit loopy. Thanks.