3.2 Matrix Algebra

1. Given the following matrices, perform the requested operation if possible.

$A=\left[\begin{array}{cc}2& 0\\ -1& 3\end{array}\right]$
$B=\left[\begin{array}{cc}4& 1\\ 2& 5\\ 0& -1\end{array}\right]$
$C=\left[\begin{array}{cc}-4& 1\\ 0& 2\end{array}\right]$
$D=\left[\begin{array}{lll}2& 1& 0\\ 0& 4& 2\end{array}\right]$
$E=\left[\begin{array}{cc}2& -3\end{array}\right]$
$F=\left[\begin{array}{c}4\\ -1\end{array}\right]$
a. $A+C$

$$\left[\begin{array}{cc}-2& 1\\ -1& 5\end{array}\right]$$

b. $3B$

$$\left[\begin{array}{cc}12& 3\\ 6& 15\\ 0& -3\end{array}\right]$$

c. $E-D$
Not possible, matrices can only be added and subtracted if they have the same dimensions

d. $AC$

$$\left[\begin{array}{cc}-8& 2\\ 4& 5\end{array}\right]$$

e. $CA$

$$\left[\begin{array}{cc}-9& 3\\ -2& 6\end{array}\right]$$

f. $AD$

$$\left[\begin{array}{ccc}4& 2& 0\\ -2& 11& 6\end{array}\right]$$

g. $DA$
Not possible, the columns of the first matrix must match the rows of the second.

h. $EF$

$$11$$

i. $FE$

$$\left[\begin{array}{cc}8& -12\\ -2& 3\end{array}\right]$$

2. Find the matrix or the composition $T\circ S$ of the linear transformations:

$$\begin{array}{c}T(\mathbf{y})=\left[\begin{array}{ccc}1& -2& 0\\ -2& 3& 1\end{array}\right]\mathbf{y}\\ S(\mathbf{x})=\left[\begin{array}{cc}2& 2\\ -1& 4\\ 0& 5\end{array}\right]\mathbf{x}\end{array}$$ $$\left[\begin{array}{cc}4& -6\\ -7& 13\end{array}\right]$$

3. Apply the row operation $R_1:R_1-3R_2$ to the matrix $A$. Show that the result is the same as $EA$ where $E$ is the elementary matrix obtained from $I_3$ by performing the same operation.

$$A=\left[\begin{array}{ccc}2& -1& 3\\ 4& -2& 1\\ 3& 4& -1\end{array}\right]$$ $$E=\left[\begin{array}{ccc}1& -3& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$$ $$EA=\left[\begin{array}{ccc}-10& 5& 0\\ 4& -2& 1\\ 3& 4& -1\end{array}\right]$$

4. Use block multiplication to find the product of $AB$.

$$A=\left[\begin{array}{rrr}1& 0& -2\\ 3& -1& 6\\ 0& 1& 2\end{array}\right],B=\left[\begin{array}{rrr}0& 2& 2\\ 1& -1& 0\\ 4& -2& 1\end{array}\right]$$

Block multiplication is supposed to make it easier, but it’s dumb. The answer is below.

5. If $A$ is a square matrix prove that $A+A^T$ is symmetric.

Adding a square matrix to it’s transposition yields a matrix which is symmetric across the diagonal because anything on one side will be added to the thing on the other side, and the sum will be on both sides in that position.

6. Show that if $A$ and $B$ are symmetric matrices and $AB=BA$ then $AB$ is also a symmetric matrix.

Using the property of transposition for matrix products, we have:

$$(AB{)}^T=B^T{A}^T.$$

Since ( A ) and ( B ) are symmetric, we substitute ( A^T = A ) and ( B^T = B ):

$$(AB{)}^T=BA.$$

Using the given condition ( AB = BA ), we substitute ( BA = AB ):

$$(AB{)}^T=AB.$$

Thus, $AB$ is symmetric, completing the proof.