1. Let $$ A=\left[\begin{array}{ccc}1 & 0 & -1 \ 0 & 1 & 0 \ 0 & -1\end{array} ight] $$ and $$ B=\left[\begin{array}{lll}1 & 1 & 2 \ 1 & 1 & 3 \ 1 & 1 & 4 \ 3 & 3 & 1\end{array} ight] $$. Compute $$ A B $$ by the four different ways I introduced. 2. Let $$ A=\left[\begin{array}{ccc}0 & -7 & -4 \ 2 & 4 & 6 \ 3 & 1 & -1\end{array} ight] $$ and $$ \boldsymbol{b}=\left[\begin{array}{c}2 \ 12 \ -2\end{array} ight] $$. Find the reduced row echelon form of the augmented matrix of the matrix equation $$ A \boldsymbol{x}=\boldsymbol{b} $$ (Hint: You can find the solution in the slide of Week 2, but I still suggest that you do it once again), and do the following: (1) Record each step (the row operation) with a matrix. (2) Suppose you get $$ t $$ matrices $$ M_{1}, M_{2}, \ldots, M_{t} $$ in order (you may name it with other names), then (a) Compute $$ E=M_{t} M_{t-1} \cdots M_{2} M_{1} $$ (Notice the order); (b) Compute $$ E A $$, and check that if it is an identity matrix. (c) Compute $$ E \boldsymbol{b} $$, and check that if it equals to the last column vector in the RREF of the augmented matrix.

Question

1. Let $$ A=\left[\begin{array}{ccc}1 & 0 & -1 \ 0 & 1 & 0 \ 0 & -1\end{array}ight] $$ and $$ B=\left[\begin{array}{lll}1 & 1 & 2 \ 1 & 1 & 3 \ 1 & 1 & 4 \ 3 & 3 & 1\end{array}ight] $$. Compute $$ A B $$ by the four different ways I introduced. 2. Let $$ A=\left[\begin{array}{ccc}0 & -7 & -4 \ 2 & 4 & 6 \ 3 & 1 & -1\end{array}ight] $$ and $$ \boldsymbol{b}=\left[\begin{array}{c}2 \ 12 \ -2\end{array}ight] $$. Find the reduced row echelon form of the augmented matrix of the matrix equation $$ A \boldsymbol{x}=\boldsymbol{b} $$ (Hint: You can find the solution in the slide of Week 2, but I still suggest that you do it once again), and do the following: (1) Record each step (the row operation) with a matrix. (2) Suppose you get $$ t $$ matrices $$ M_{1}, M_{2}, \ldots, M_{t} $$ in order (you may name it with other names), then (a) Compute $$ E=M_{t} M_{t-1} \cdots M_{2} M_{1} $$ (Notice the order); (b) Compute $$ E A $$, and check that if it is an identity matrix. (c) Compute $$ E \boldsymbol{b} $$, and check that if it equals to the last column vector in the RREF of the augmented matrix.

Greene Marshall · Accepted Answer

For Problem 1, the multiplication $$ A B $$ is not defined due to the mismatch in dimensions. For Problem 2, the reduced row echelon form of the augmented matrix is: $$ \begin{bmatrix} 1 & 0 & 0 & | & \frac{31}{7} \ 0 & 1 & 0 & | & -\frac{10}{7} \ 0 & 0 & 1 & | & 1 \end{bmatrix} $$ The steps to achieve this form involve row swaps, scaling, and elimination operations.

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