1 Matrices
Section 1 The Basic Operations
1.1. What are the entries \( a_{21} \), and \( a_{23} \) of the matrix A = \( \begin{bmatrix}
1 &2 &5 \\
2 &7 &8 \\
0 &9 &4
\end{bmatrix} \) ?
\( a_{21} = 2 \), and \( a_{23} = 8 \)
1.2. Determine the products AB and BA for the following values of A and B:
\( A = \begin{bmatrix}
1 &2 &3 \\
3 &3 &1
\end{bmatrix}, \hspace{2mm}
B = \begin{bmatrix}
-8 &-4 \\
9 &5 \\
-3 &-2
\end{bmatrix} \)
\( AB = \begin{bmatrix}
1 &0 \\
0 &1
\end{bmatrix}, \hspace{4mm}
BA = \begin{bmatrix}
-20 &-28 &-28 \\
24 &33 &32 \\
-9 &-12 &-11
\end{bmatrix} \)
\(
A = \begin{bmatrix}
1 &4 \\
1 &2
\end{bmatrix}, \hspace{2mm}
B = \begin{bmatrix}
6 &-4 \\
3 &2
\end{bmatrix}
\)
\( AB = \begin{bmatrix}
18 &4 \\
12 &0
\end{bmatrix}, \hspace{4mm}
BA = \begin{bmatrix}
2 &16 \\
5 &16
\end{bmatrix} \)
1.4. Verify the associative law for the matrix product
\( ABC = \begin{bmatrix}
1 &2 \\
0 &1
\end{bmatrix}
\begin{bmatrix}
0 &1 &2 \\
1 &1 &3
\end{bmatrix}
\begin{bmatrix}
1 \\
4 \\
3
\end{bmatrix} \)
Note: This is a self-checkingproblem.It won’t come out unless you multiply correctly. If you need to practice matrix multiplication, use this problem as a model.
\( (AB)C = \begin{bmatrix}
2 &3 &8 \\
1 &1 &3
\end{bmatrix}
\begin{bmatrix}
1 \\
4 \\
3
\end{bmatrix}
= \begin{bmatrix}
38 \\
14
\end{bmatrix}
= 2 \begin{bmatrix}
19 \\
7
\end{bmatrix}
\)
\( A(BC) = \begin{bmatrix}
1 &2 \\
0 &1
\end{bmatrix}
\begin{bmatrix}
10 \\
14
\end{bmatrix}
= \begin{bmatrix}
38 \\
14
\end{bmatrix}
= 2 \begin{bmatrix}
19 \\
7
\end{bmatrix}
\)