Exercises: Represent Systems of Linear Equations as Matrix Equations

The matrix $A = \begin{bmatrix} 4 & -1 \\ 0 & 7 \end{bmatrix}$ is a matrix of dimensions:

For the matrix $B = \begin{bmatrix} 5 & 3 \\ -2 & 8 \end{bmatrix}$, what is the entry $b_{12}$?

Which of the following correctly computes the $(1, 1)$ entry of the product $\begin{bmatrix} 2 & 3 \end{bmatrix} \cdot \begin{bmatrix} 4 \\ 1 \end{bmatrix}$?

Compute the matrix product $\begin{bmatrix} 2 & 1 \\ 3 & -4 \end{bmatrix} \begin{bmatrix} 5 & 0 \\ -1 & 2 \end{bmatrix}$. Show each entry as a dot product.

Compute $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} 3 & -2 \\ 5 & 7 \end{bmatrix}$. What do you notice about the result?

Let $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ and $B = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$.

Compute both $AB$ and $BA$. Are they equal?

Write the system $x + 3y = 5$ and $2x - y = 1$ as a matrix equation $A\mathbf{x} = \mathbf{b}$. Identify $A$, $\mathbf{x}$, and $\mathbf{b}$. Then verify by computing $A\mathbf{x}$ and confirming it equals the left-hand sides of the system.

The matrix equation $\begin{bmatrix} 3 & 1 \\ -2 & 4 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 6 \\ 8 \end{bmatrix}$ represents a system of two equations. Write out the system.