Problems

01

Basic multiplications; differing sizes

Define the following matrices:

$$A=\left(\begin{array}{cc}1& 2\\ 3& 6\\ 2& 1\end{array}\right),B=\left(\begin{array}{cc}1& -1\\ -1& 0\end{array}\right),C=\left(\begin{array}{ccc}1& 0& 1\\ 1& 3& 3\end{array}\right).$$

Compute the matrix products: $AB$, $AC$, $(BC)A$, $B(CA)$.

02

Row reduction using elementary matrices

The row-scale, row-add, and row-swap matrices can be applied in a sequence to manipulate the entries of another matrix or vector. A very useful goal is to convert a matrix into an upper-triangular matrix, which is a matrix that has all entries below the main diagonal equal to zero.

For example, the matrix $\left(\begin{array}{cc}2& -4\\ 3& -1\end{array}\right)$ can be manipulated into an upper-triangular matrix by adding $-3/2$ times row $1$ into row $2$, because $(-3/2)\cdot 2=-3$ and $-3+3=0$. This action is performed by the row-add matrix with $-3/2$ in the $2^{\text{nd}}$ row of the $1^{\text{st}}$ column:

$$\left(\begin{array}{cc}1& 0\\ -3/2& 1\end{array}\right)\left(\begin{array}{cc}2& -4\\ 3& -1\end{array}\right)=\left(\begin{array}{cc}2& -4\\ 0& 5\end{array}\right).$$

Now, using a sequence of matrix actions chosen from the types row-scale and row-add, convert the matrix

$$A=\left(\begin{array}{ccc}3& 2& -2\\ 15& 12& -8\\ 9& 2& -6\end{array}\right)$$

into an upper-triangular matrix. (Hint: first create two zeros in the first column, then create the zero needed in the second column.)

Now multiply together the manipulation matrices used to perform your actions (multiplied right-to-left in order of the actions so that the actions are composed).

Finally, show that this product matrix converts the original matrix to an upper-triangular matrix by multiplying on the left.

03

Matrix inversion to find preimages

In each case below, show that $\det A\ne 0$, compute the inverse $A^{-1}$, and use it to find the vector $\mathbf{u}$ satisfying the equation:

• (a) $\left(\begin{array}{cc}1& 2\\ 3& 5\end{array}\right)\mathbf{u}=\left(\begin{array}{c}5\\ -2\end{array}\right)$
• (b) $\left(\begin{array}{cc}2& 5\\ -3& -7\end{array}\right)\mathbf{u}=\left(\begin{array}{c}1\\ 8\end{array}\right)$

04

Calculate eigenvalues and eigenvectors for $2\times 2$ matrix

Imitate the Example to find the eigenvalues and eigenvectors for $A$:

$$A=\left(\begin{array}{cc}7& 4\\ -3& -1\end{array}\right).$$

05

Actions on the right: rows v. columns

In this problem, we consider the action of a matrix upon another matrix by multiplication on the right. That is, the action of $B$ upon $A$ by sending $A$ to $AB$. (The effect is very similar to acting on the left, except that the roles of ‘rows’ and ‘columns’ are reversed!)

• (a) Describe the action of a $3\times 3$ matrix $B$ on the right upon a row vector $(v_1{v}_2{v}_3)$. (To obtain the formula, take the matrix product treating the row vector as a $1\times 3$ matrix. Now interpret the formula analogously to the first point in the Summary.)
• (b) Describe the action of $B$ multiplying on the right on a matrix $A$ in terms of the row vectors of $A$.
• (c) What does the matrix $B_3(\lambda )=\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& \lambda \end{array}\right)$ do to another $3\times 3$ matrix $A$ when it acts on the right?
• (d) What is the $3\times 3$ matrix that swaps columns $i$ and $j$ by acting on the right?
• (e) What is the $3\times 3$ matrix that adds column $j$ into column $i$ by acting on the right?