Learn Matrices: Practice Problems and Examples

Let
$\left[\begin{array}{c}3\\ 1\end{array}\right]$
and $\left[\begin{array}{c}8\\ 6\end{array}\right]$
Compute the distance from y to the line through u and the origin.

Determine if b is a linear combination of the vectors formed from the columns of the matrix A.
$A=\left[\begin{array}{ccc}1& -4& 2\\ 0& 3& 5\\ -2& 8& -4\end{array}\right],b=\left[\begin{array}{c}3\\ -7\\ -3\end{array}\right]$

Find the quadratic function that is the best fit for​ f(x) defined by the table below x f(x) 0 1 2 -0.6 4 -1.4 6 -1.4

Show that if A ⊆ C and B ⊆ D, then A × B ⊆ C × D

The owner of a grocery store can sell 2654 gallons of milk each week at $2.41 per gallon and 2954 gallons of milk each week when he offers a sale price of $1.91 per gallon. Assume that the number of gallons sold S is a linear function of the price p per gallon.
a) Find a formula for the gallon sales, S(p), as a function of price p. (Don't include S in your answer.)
b) How many gallons would you expect to sell at $3.21 per gallon?

Find the cross product ${\displaystyle a\times b}$ and verify that it is orthogonal to both a and b. ${\displaystyle a=⟨6,0,-2⟩,b=⟨0,8,0⟩}$

The length of a rectangle is 55 meters less than 33 times the width. The perimeter is 14 meters. Find the width.

As a test of orienteering skills, your physics class holds a con test in a large, open field. Each contestant is told to travel 20.8 m due north from the starting point, then 38.0 m due east, and finally 18.0 m in the direction 33.0° west of south. After the specified displacements, a contestant will find a silver dollar hidden under a rock. The winner is the person who takes the shortest time to reach the location of the silver dollar. Remembering what you learned in class, you run on a straight line from the starting point to the hidden coin. How far and in what direction do you run?

Construct a ${\displaystyle 3\times 3}$ matrix A, with nonzero entries, and a vector b in ${\displaystyle {\mathbb{R}}^3}$ such that b is not in the set spanned by the columns of A.

Construct a ${\displaystyle 3\times 3}$ matrix, not in echelon form, whose columns span ${\displaystyle R^3}$. Show that the matrix you construct has the desired property.

For the matrices, list the eigenvalues, repeated according to their multiplicities.
$\left[\begin{array}{cccc}4& -7& 0& 2\\ 0& 3& -4& 6\\ 0& 0& 3& -8\\ 0& 0& 0& 1\end{array}\right]$

Find the determinant of the matrix:
${\displaystyle \left[\begin{array}{ccc}2& 1& 4\\ -3& 1& 5\\ 4& 0& 0\end{array}\right]}$

Let ${\displaystyle A=\left[\begin{array}{ccc}4& 0& 5\\ -1& 3& 2\end{array}\right]}$
${\displaystyle B=\left[\begin{array}{ccc}1& 1& 1\\ 3& 5& 7\end{array}\right]}$
${\displaystyle C=\left[\begin{array}{ccc}2& & -3\\ 0& & 1\end{array}\right]}$
Find ${\displaystyle 3A–B;C\cdot B+A.}$

Tell which of the following subsets are subspaces of the vector space of nxn matrices, ${\displaystyle M_{nxn}}$
a) The set of nxn matrices with a zero determinant.
b) The invertible matrices.
c) The diagonal matrices.
d) Upper triangular matrices.
e) The nxn matrices, all of ${\displaystyle M_{nxn}}$.

 

If the roots of $a{x}^2+bx+c=0$ are $\frac{3}{2},\frac{4}{3}$ then $(a+b+c{)}^2=$

 

For each topic, decide how a scatterplot of the data would likely look. Explain your reasoning. Speed of a runner and amount of time to complete a race

For the following exercises, enter the data from each table into a graphing calculator and graph the resulting scatter plots. Determine whether the data from the table could represent a function that is linear, exponential, or logarithmic. $\begin{array}{|ccccccccccc|}\hline x& 1& 2& 3& 4& 5& 6& 7& 8& 9& 10\\ f(x)& 2.4& 2.88& 3.456& 4.147& 4.977& 5.972& 7.166& 8.6& 10.32& 12.383\\ \hline\end{array}$

Use the definition of continuity and the properties of limits to show that the function is continuous on the given interval.
${\displaystyle g\left(x\right)=2\sqrt{3}-x,(-\mathrm{\infty },3]}$

Substitution and elimination, and matrix methods such as the Gauss-Jordan method and Cramer's rule. Use each method at least once when solving the systems below. include solutions with nonreal complex number components. For systems with infinitely many solutions, write the solution set using an arbitrary variable.
$2x+3y+4z=3$
$-4x+2y-6z=2$
$4x+3z=0$

Transform the given differential equation or system into an equivalent system of first-order differential equations.