Linear equations

Study guide and 10 practice problems on:

To analyze a system of linear equations, it is convenient to put it into the matrix form $$\mathbf{A} \mathbf{x} = \mathbf{b},$$ where $\mathbf{A}$ is a known matrix, $\mathbf{b}$ is a known column vector, and $\mathbf{x}$ is a column vector of unknowns.

If $\bfA$ is square and invertible, the solution to $\bfA \bfx = \bfb$ is $\bfx = \bfA^{-1} \bfb.$

If $\mathbf{b}=0$, then the equation is called homogeneous.

If $\mathbf{b} \neq 0$, then the equation is called inhomogeneous.

Study Guide

Inhomogeneous linear equations
(2 problems)
Homogeneous linear equations
(3 problems)

Practice problems

For what value of $c$ is there a nonzero solution to the following equation? For that value of $c$, find all solutions to the equation. $$\begin{pmatrix}1&1\\2&c\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix} = \begin{pmatrix}0\\0\end{pmatrix}$$
Solution
For what values of $\lambda$ are there nontrivial solutions to $$\begin{pmatrix}1&0&0\\0&2&0\\0&0&3\end{pmatrix}\begin{pmatrix}x\\y\\z\end{pmatrix} = \lambda \begin{pmatrix}x\\y\\z\end{pmatrix}$$
For each value, find a nontrivial solution.
Solution
Are there any real values of $c$ for which there is a nontrivial (nonzero) solution to $$\begin{pmatrix}1&c\\-c&2\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix} = \begin{pmatrix}0\\0\end{pmatrix}?$$
Solution
How many solutions are there to $$\begin{pmatrix}1&1&1\\1&1&0\\0&0&1\end{pmatrix}\begin{pmatrix}x\\y\\z\end{pmatrix} = \begin{pmatrix}1\\2\\3\end{pmatrix}?$$
Solution
How many solutions are there to $$\begin{pmatrix}1&1&1\\1&1&0\\0&0&1\end{pmatrix}\begin{pmatrix}x\\y\\z\end{pmatrix} = \begin{pmatrix}3\\2\\1\end{pmatrix}?$$ If there are any, find all of them.
Solution
Write this matrix equation as a system of 3 equations. Solve for $x,y,z$: $$\begin{pmatrix}1 & 1& 1\\0 & 1 & 1\\ 0 &0 & 1 \end{pmatrix} \begin{pmatrix}x\\y\\z\end{pmatrix} = \begin{pmatrix}4\\3\\1\end{pmatrix}$$
Solution
Find the second degree polynomial going through $(-1, 1), (1, 3),$ and $(2,2)$.

Hint: To find the coefficients of $y = a + bx + cx^2$, set up a $3 \times 3$ matrix satisfied by $a,b,c$.
Solution
Write the following system as a matrix equation for $x,y,z$:\begin{align}
y +z &=4\\
2x -y &=z
\end{align}
Solution
Find $\bfA^{-1}$ and use it to solve $\bfA \bfx = \bfb$, where $$\bfA = \begin{pmatrix}1 & -1& 1 \\ 1 & 1 & 1\\1 & 2 & 4 \end{pmatrix}, \quad \bfb = \begin{pmatrix} 1\\3\\2 \end{pmatrix}$$
Solution
Solve by matrix inversion: $$\begin{pmatrix} 2 & 3 \\ 10 & 16 \end{pmatrix} \begin{pmatrix} x\\y \end{pmatrix} = \begin{pmatrix}1\\2\end{pmatrix}.$$
Solution

Linear equations

Study Guide

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