Learn Matrices: Practice Problems and Examples

One cake recipe calls for ${\displaystyle \frac{2}{3}}$ cup of sugar. Another recipe calls for ${\displaystyle 1\frac{1}{4}}$ cups of sugar. How many cups of sugar are needed to make both cakes?

Tooth decay occurs when tooth enamel wears away over time. It exponential decay models the change in the number of dentists in the United States over time, do you think the number of dentists in the United States is increasing or decreasing?

A motorcycle is traveling at 42 miles per hour on tires that measure 1.2 feet in diameter. Find the approximate angular speed of the tires in radians per minute.

A scatterplot shows a moderately strong negative linear relationship. Use your calculator or software to find the equation of the least-squares regression line.

A chef has one cheese that contains 45% fat and another cheese that contains 20% fat. How many grams of each cheese should she use in order to obtain 30 grams of a cheese mixture that is 30% fat?

Penguins spend almost 75% of their lives in the sea. An Emperor Penguin in the wild has a life span of about 18 years. About how many years does this penguin spend In the sea?

A matrix is defined as a rectangular array of numbers. Many of the arrays that you have worked have additional features, such as labels on the rows and columns or entries that are numbers with units. An array with additional features is technically not a matrix, instead we would call it a table. The difference between a matrix and a table is not vitally important in this unit, so we have worked with the underlying array of numbers and not worried about this technical detail. In more advanced courses in mathematics, like Linear Algebra, it will be important to be very precise about matrices. For now, describe and give examples of some of the ways a table can be different from a matrix. In your description and examples, be sure to include differences related to at least these three characteristics: entries, labels, and operations.

How close to 0 do we need to take x so that
$(x^2+6x+9)<9.001?$

One of the five given matrices represents an orthogonal projection onto a line and another represents a reflection about line. Identify both and briefly justify your choice.

Explain the term Comparable matrices?

Let A and B be similar $n\times n$ matrices. Prove that if A is idempotent, then B is idempotent. (X is idempotent if $X^2=X$ )

Determine , if possible , the value of x which will male the following true:
$\left[\begin{array}{cc}4& 3\\ 1& x\end{array}\right]+2\left[\begin{array}{c}2\\ 1\end{array}\right]\left[\begin{array}{cc}3& x\end{array}\right]=\left[\begin{array}{cc}16& 11\\ 7& 6\end{array}\right]$

Let a linear sytem of equations Ax=b where
$A=\left(\begin{array}{ccc}4& 2& -2\\ 2& 2& -3\\ -2& -3& 14\end{array}\right),b={\left(\begin{array}{c}10,5,4\end{array}\right)}^T$
in case we solve this equation system by using Dolittle LU factorization method , find Z and X matrices

Find three $2\times 2$ idempotent matrices. (Recall that a square matrix A is idempotent when $A^2=A$)

Use cramer's rule to determine the values of $I_1,I_2,I_3$ and $I_4$
$\left[\begin{array}{cccc}13.7& -4.7& -2.2& 0\\ -4.7& 15.4& 0& -8.2\\ -2.2& 0& 25.4& -22\\ 0& -8.2& -22& 31.3\end{array}\right]\left[\begin{array}{c}{I}_1\\ I_2\\ I_3\\ I_4\end{array}\right]=\left[\begin{array}{c}6\\ -6\\ 5\\ -9\end{array}\right]$

Write each of the following linear combinations of columns as a linear system of the form in (4):
$a)x_1\left[\begin{array}{c}2\\ 0\end{array}\right]+x_2\left[\begin{array}{c}1\\ 3\end{array}\right]=\left[\begin{array}{c}4\\ 2\end{array}\right]$
$b)x_1\left[\begin{array}{c}1\\ 2\\ -1\end{array}\right]+x_2\left[\begin{array}{c}0\\ 1\\ 2\end{array}\right]+x_3\left[\begin{array}{c}3\\ 4\\ 5\end{array}\right]+x_4\left[\begin{array}{c}1\\ 3\\ 4\end{array}\right]=\left[\begin{array}{c}2\\ 5\\ 8\end{array}\right]$

Perform the indicated matrix operations:
$\left[\begin{array}{cc}5& 4\\ -3& 7\\ 0& 1\end{array}\right]-\left[\begin{array}{cc}-4& 8\\ 6& 0\\ -5& 3\end{array}\right]$

Given matrix A and matrix B. Find (if possible) the matrices: (a) AB (b) BA.
$A=\left[\begin{array}{ccc}1& -1& 1\\ 5& 0& -2\\ 3& -2& 2\end{array}\right],B=\left[\begin{array}{ccc}1& 1& 0\\ 1& -4& 5\\ 3& -1& 2\end{array}\right]$

Use the graphing calculator to solve if possible
$$\begin{gathered} A=\left[\begin{array}{ccc}1& 0& 5\\ 1& -5& 7\\ 0& 3& -4\end{array}\right] \\ B=\left[\begin{array}{ccc}3& -5& 3\\ 2& 3& 1\\ 4& 1& -3\end{array}\right] \\ C=\left[\begin{array}{ccc}5& 2& 3\\ 2& -1& 0\end{array}\right] \\ D=\left[\begin{array}{c}5\\ -3\\ 4\end{array}\right] \end{gathered}$$
Find the value in row 2 column 3 of $AB-3B$

(square roots of the identity matrix) For how many $2\times 2$ matrices A is it true that $A^2=I$ ? Now answer the same question for $n\times x$ n matrices where $n>2$