Question

Consider the matrix \[E=\left[\begin{array}{rrr} 1 & 0 & 0 \\ -3 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]\] and an arbitrary $3 \times 3$ matrix \[A=\left[\begin{array}{lll} a & b & c \\ d & e & f \\ g & h & k \end{array}\right].\] a. Compute $E A$. Comment on the relationship between $A$ and $E A$, in terms of the technique of elimination we learned in Section 1.2. b. Consider the matrix \[E=\left[\begin{array}{lll} 1 & 0 & 0 \\ 0 & \frac{1}{4} & 0 \\ 0 & 0 & 1 \end{array}\right]\] and an arbitrary $3 \times 3$ matrix $A$. Compute $E A$. Comment on the relationship between $A$ and $E A$ c. Can you think of a $3 \times 3$ matrix $E$ such that $E A$ is obtained from $A$ by swapping the last two rows (for any $3 \times 3$ matrix $A$ )? d. The matrices of the forms introduced in parts (a), (b), and (c) are called elementary: An $n \times n$ matrix $E$ is elementary if it can be obtained from $I_{n}$ by performing one of the three elementary row operations on $I_{n} .$ Describe the format of the three types of elementary matrices.

          Consider the matrix
\[E=\left[\begin{array}{rrr}
1 & 0 & 0 \\
-3 & 1 & 0 \\
0 & 0 & 1
\end{array}\right]\]
and an arbitrary $3 \times 3$ matrix
\[A=\left[\begin{array}{lll}
a & b & c \\
d & e & f \\
g & h & k
\end{array}\right].\]
a. Compute $E A$. Comment on the relationship between $A$ and $E A$, in terms of the technique of elimination we learned in Section 1.2.
b. Consider the matrix
\[E=\left[\begin{array}{lll}
1 & 0 & 0 \\
0 & \frac{1}{4} & 0 \\
0 & 0 & 1
\end{array}\right]\]
and an arbitrary $3 \times 3$ matrix $A$. Compute $E A$. Comment on the relationship between $A$ and $E A$
c. Can you think of a $3 \times 3$ matrix $E$ such that $E A$ is obtained from $A$ by swapping the last two rows (for any $3 \times 3$ matrix $A$ )?
d. The matrices of the forms introduced in parts (a),
(b), and (c) are called elementary: An $n \times n$ matrix $E$ is elementary if it can be obtained from $I_{n}$ by performing one of the three elementary row operations on $I_{n} .$ Describe the format of the three types of elementary matrices.

Added by Charles H.

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