Multivariable calculus and vector analysis

Matrix and vector multiplication examples

Example 1

Compute $A\mathbf{x}$ where $\mathbf{x}=\left(-2,-1,0\right)$

$$\begin{array}{lll}\hfill A=\left[\begin{array}{ccc}\hfill 1& \hfill 2& \hfill 3\\ \hfill 4& \hfill 5& \hfill 6\\ \hfill 7& \hfill 8& \hfill 9\\ \hfill 10& \hfill 11& \hfill 12\end{array}\right].& & \hfill \end{array}$$

Solution:

$$\begin{array}{llll}\hfill A\mathbf{x}& =\left[\begin{array}{ccc}\hfill 1& \hfill 2& \hfill 3\\ \hfill 4& \hfill 5& \hfill 6\\ \hfill 7& \hfill 8& \hfill 9\\ \hfill 10& \hfill 11& \hfill 12\end{array}\right]\left[\begin{array}{c}\hfill -2\\ \hfill 1\\ \hfill 0\end{array}\right]& \hfill & \\ \hfill & =\left[\begin{array}{c}\hfill -2\cdot 1+1\cdot 2+0\cdot 3\hfill \\ \hfill -2\cdot 4+1\cdot 5+0\cdot 6\hfill \\ \hfill -2\cdot 7+1\cdot 8+0\cdot 9\hfill \\ \hfill -2\cdot 10+1\cdot 11+0\cdot 12\hfill \end{array}\right]& \hfill & \\ \hfill & =\left[\begin{array}{c}\hfill 0\hfill \\ \hfill -3\hfill \\ \hfill -6\hfill \\ \hfill -9\hfill \end{array}\right]=\left(0,-3,-6,-6\right).& \hfill & \end{array}$$

Example 2

Compute $A\mathbf{y}$ where $\mathbf{y}=\left(-3,-2,-1,0\right)$ and $A$ is as in Example 1.

Solution: The matrix-vector product is not defined. $A$ is $4\times 3$ and $\mathbf{y}$ is $4\times 1$ (viewed as column vector).

Example 3

Compute $BC$, where

$$\begin{array}{lll}\hfill B=\left[\begin{array}{ccc}\hfill 1& \hfill 2& \hfill 3\\ \hfill 4& \hfill 5& \hfill 6\end{array}\right]\text{ and}C=\left[\begin{array}{cc}\hfill 1& \hfill 2\\ \hfill 3& \hfill 4\\ \hfill 5& \hfill 6\end{array}\right].& & \hfill \end{array}$$

Solution:

$$\begin{array}{llll}\hfill BC& =\left[\begin{array}{ccc}\hfill 1& \hfill 2& \hfill 3\\ \hfill 4& \hfill 5& \hfill 6\end{array}\right]\left[\begin{array}{cc}\hfill 1& \hfill 2\\ \hfill 3& \hfill 4\\ \hfill 5& \hfill 6\end{array}\right]& \hfill & \\ \hfill & =\left[\begin{array}{ccc}\hfill 1\cdot 1+2\cdot 3+3\cdot 5\hfill & \hfill \hfill & \hfill 1\cdot 2+2\cdot 4+3\cdot 6\hfill \\ \hfill 4\cdot 1+5\cdot 3+6\cdot 5\hfill & \hfill \hfill & \hfill 4\cdot 2+5\cdot 4+6\cdot 6\hfill \end{array}\right]& \hfill & \\ \hfill & =\left[\begin{array}{cc}\hfill 22\hfill & \hfill 28\hfill \\ \hfill 49\hfill & \hfill 64\hfill \end{array}\right]& \hfill & \end{array}$$

Example 4

Using $B$ and $C$ as defined in Example 3, calculate $CB$.

Solution:

$$\begin{array}{llll}\hfill CB& =\left[\begin{array}{cc}\hfill 1& \hfill 2\\ \hfill 3& \hfill 4\\ \hfill 5& \hfill 6\end{array}\right]\left[\begin{array}{ccc}\hfill 1& \hfill 2& \hfill 3\\ \hfill 4& \hfill 5& \hfill 6\end{array}\right]& \hfill & \\ \hfill & =\left[\begin{array}{ccccc}\hfill 1\cdot 1+2\cdot 4\hfill & \hfill \hfill & \hfill 1\cdot 2+2\cdot 5\hfill & \hfill \hfill & \hfill 1\cdot 3+2\cdot 6\hfill \\ \hfill 3\cdot 1+4\cdot 4\hfill & \hfill \hfill & \hfill 3\cdot 2+4\cdot 5\hfill & \hfill \hfill & \hfill 3\cdot 3+4\cdot 6\hfill \\ \hfill 5\cdot 1+6\cdot 4\hfill & \hfill \hfill & \hfill 5\cdot 2+6\cdot 5\hfill & \hfill \hfill & \hfill 5\cdot 3+6\cdot 6\hfill \end{array}\right]& \hfill & \\ \hfill & =\left[\begin{array}{ccc}\hfill 9& \hfill 12& \hfill 15\\ \hfill 19& \hfill 26& \hfill 33\\ \hfill 29& \hfill 40& \hfill 51\end{array}\right]& \hfill & \end{array}$$

Clearly, one can see that matrix multiplication is not commutative, i.e., $BC\ne CB$. In the case of examples 3 and 4, $BC$ isn’t even the same size matrix as $CB$. In some other cases, $BC$ might be defined but $CB$ won’t be defined (for example, when $B$ is a $3\times 2$ matrix and $C$ is a $2\times 4$ matrix). It is even true that when $B$ and $C$ are square matrices, matrix multiplication is not commutative. You can try yourself and see that $BC\ne CB$ if

$$\begin{array}{lll}\hfill B=\left[\begin{array}{cc}\hfill 1& \hfill 2\\ \hfill 3& \hfill 4\end{array}\right]\text{ and}C=\left[\begin{array}{cc}\hfill 5& \hfill 6\\ \hfill 7& \hfill 8\end{array}\right].& & \hfill \end{array}$$

Contents by topic

1. Geometry in higher dimensions

1a. Preliminaries

1b. Lines and planes

1c. Matrices

• Matrices and determinants
• Multiplying matrices and vectors
• Dot product in matrix notation
• Matrices and linear functions
• The relationship between determinants and area or volume
• Matrix and vector multiplication examples

1d. Visualizing functions in higher dimensions

2. Differential Calculus

3. Integral calculus

4. Vector calculus

5. The fundamental theorems of vector calculus

6. Review material