2.4: Review Problems

1. While performing Gaussian elimination on these augmented matrices write the full system of equations describing the new rows in terms of the old rows above each equivalence symbol as in Example 20.

$$\left(\begin{array}{rr|r} 2 & 2 & 10 \\ 1 & 2 & 8 \\ \end{array}\right) ,~ \left(\begin{array}{rrr|r} 1 & 1 & 0 & 5 \\ 1 & 1 & -1& 11 \\ -1 & 1 & 1 & -5 \\ \end{array}\right)$$

2. Solve the vector equation by applying ERO matrices to each side of the equation to perform elimination. Show each matrix explicitly as in Example 23.

$$\begin{eqnarray*} \begin{pmatrix} 3 &6 &2 \\ -3 5 &9 &4 \\ 1 2 &4 &2 \\ 0 \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} -3 \\ 1 \\ 0 \\ \end{pmatrix} \end{eqnarray*}$$

3. Solve this vector equation by finding the inverse of the matrix through $(M|I)\sim (I|M^{-1})$ and then applying $M^{-1}$ to both sides of the equation.

$$\begin{eqnarray*} \begin{pmatrix} 2 &1 &1 \\ 9 1 &1 &1 \\ 6 1 &1 &2 \\ 7 \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 9 \\ 6 \\ 7 \\ \end{pmatrix} \end{eqnarray*}$$

4. Follow the method of Examples 28 and 29 to find the $LU$ and $LDU$ factorization of

$$\begin{eqnarray*} \begin{pmatrix} 3 &3 &6 \\ 3 &5 &2 \\ 6 &2 &5 \\ \end{pmatrix} \end{eqnarray*}$$

5. Multiple matrix equations with the same matrix can be solved simultaneously.

a) Solve both systems by performing elimination on just one augmented matrix.

$$\begin{eqnarray*} \begin{pmatrix} 2 &-1 &-1 \\ -1 &1 &1 \\ 1 &-1 &0 \\ \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 0\\ 1 \\ 0 \\ \end{pmatrix} ,~ \begin{pmatrix} 2 &-1 &-1 \\ -1 &1 &1 \\ 1 &-1 &0 \\ \end{pmatrix} \begin{pmatrix} a \\ b \\ c \end{pmatrix} = \begin{pmatrix} 2\\ 1 \\ 1 \\ \end{pmatrix} \end{eqnarray*}$$

b) What are the columns of $M^{-1}$ in $(M|I)\sim (I|M^{-1})$?

6. How can you convince your fellow students to never make this mistake?

$$\begin{eqnarray*} \left(\begin{array}{rrr|r} 1 & 0 & 2 & 3 \\ 0 & 1 & 2& 3 \\ 2 & 0 & 1 & 4 \\ \end{array}\right) & \stackrel{R_1'=R_1+R_2}{ \stackrel{R_2'=R_1-R_2}{ \stackrel{\ R_3'= R_1+2R_2}{\sim}}} & \left(\begin{array}{rrr|r} 1 & 1 & 4 & 6 \\ 1 & -1 & 0& 0 \\ 1 & 2 & 6 & 9 \end{array}\right) \end{eqnarray*}$$

7. Is $LU$ factorization of a matrix unique? Justify your answer.
$\infty$. If you randomly create a matrix by picking numbers out of the blue, it will probably be difficult to perform elimination or factorization; fractions and large numbers will probably be involved. To invent simple problems it is better to start with a simple answer:

1. Start with any augmented matrix in RREF. Perform EROs to make most of the components non-zero. Write the result on a separate piece of paper and give it to your friend. Ask that friend to find RREF of the augmented matrix you gave them. Make sure they get the same augmented matrix you started with.
2. Create an upper triangular matrix $U$ and a lower triangular matrix $L$ with only $1$s on the diagonal. Give the result to a friend to factor into $LU$ form.
3. Do the same with an $LDU$ factorization.

Contributor

• David Cherney, Tom Denton, and Andrew Waldron (UC Davis)