\(\text{www.the}\beta\text{etafunction.com}\)
Home
Login
Questions: Algebra BusinessCalculus
Please login to create an exam or a quiz.
Use the matrices below to answer the following questions: \begin{equation*}A = \left[\begin{matrix}-3 & 0\\-12 & 5\\14 & 3\end{matrix}\right] B = \left[\begin{matrix}-4 & -13 & -4\\3 & -12 & -15\end{matrix}\right] C = \left[\begin{matrix}15 & 7\end{matrix}\right] D = \left[\begin{matrix}7 & 8 & -7\\5 & 5 & 2\\-2 & -2 & -1\end{matrix}\right] E = \left[\begin{matrix}15 & -12\\3 & -18\\17 & 6\end{matrix}\right] \end{equation*}
\begin{question}Use the matrices below to answer the following questions:
\begin{equation*}A = \left[\begin{matrix}-3 & 0\\-12 & 5\\14 & 3\end{matrix}\right] B = \left[\begin{matrix}-4 & -13 & -4\\3 & -12 & -15\end{matrix}\right] C = \left[\begin{matrix}15 & 7\end{matrix}\right] D = \left[\begin{matrix}7 & 8 & -7\\5 & 5 & 2\\-2 & -2 & -1\end{matrix}\right] E = \left[\begin{matrix}15 & -12\\3 & -18\\17 & 6\end{matrix}\right] \end{equation*}
\begin{enumerate}
\item (10pts) Find $A + E$ and $C + D$
\soln{9cm}{
$\left[\begin{matrix}-3 & 0\\-12 & 5\\14 & 3\end{matrix}\right]+\left[\begin{matrix}15 & -12\\3 & -18\\17 & 6\end{matrix}\right]=\left[\begin{matrix}12 & -12\\-9 & -13\\31 & 9\end{matrix}\right]$ The sum is undefined. The matricies do not have the same shape.
}
\item (10pts) Find $BC$ and $CB$
\soln{9cm}{
The product is undefined. $\left[\begin{matrix}15 & 7\end{matrix}\right]\left[\begin{matrix}-4 & -13 & -4\\3 & -12 & -15\end{matrix}\right]=\left[\begin{matrix}-39 & -279 & -165\end{matrix}\right]$
}
\item (10pts) Find the inverse of Matrix D, that is $D^{-1}$
\soln{9cm}{
$\left[\begin{matrix}-1 & 22 & 51\\1 & -21 & -49\\0 & -2 & -5\end{matrix}\right]$
}
\end{enumerate}
\end{question}
\documentclass{article}
\usepackage{tikz}
\usepackage{amsmath}
\usepackage[margin=2cm]{geometry}
\usepackage{tcolorbox}
\newcounter{ExamNumber}
\newcounter{questioncount}
\stepcounter{questioncount}
\newenvironment{question}{{\noindent\bfseries Question \arabic{questioncount}.}}{\stepcounter{questioncount}}
\renewcommand{\labelenumi}{{\bfseries (\alph{enumi})}}
\newif\ifShowSolution
\newcommand{\soln}[2]{%
\ifShowSolution%
\noindent\begin{tcolorbox}[colframe=blue,title=Solution]#2\end{tcolorbox}\else%
\vspace{#1}%
\fi%
}%
\newcommand{\hideifShowSolution}[1]{%
\ifShowSolution%
%
\else%
#1%
\fi%
}%
\everymath{\displaystyle}
\ShowSolutiontrue
\begin{document}\begin{question}(10pts) The question goes here!
\soln{9cm}{The solution goes here.}
\end{question}\end{document}<p> <p>Use the matrices below to answer the following questions:
<img class="equation_image" title=" A = \left[\begin{matrix}-3 & 0\\-12 & 5\\14 & 3\end{matrix}\right] B = \left[\begin{matrix}-4 & -13 & -4\\3 & -12 & -15\end{matrix}\right] C = \left[\begin{matrix}15 & 7\end{matrix}\right] D = \left[\begin{matrix}7 & 8 & -7\\5 & 5 & 2\\-2 & -2 & -1\end{matrix}\right] E = \left[\begin{matrix}15 & -12\\3 & -18\\17 & 6\end{matrix}\right] " src="/equation_images/%20A%20%3D%20%5Cleft%5B%5Cbegin%7Bmatrix%7D-3%20%26%200%5C%5C-12%20%26%205%5C%5C14%20%26%203%5Cend%7Bmatrix%7D%5Cright%5D%20B%20%3D%20%5Cleft%5B%5Cbegin%7Bmatrix%7D-4%20%26%20-13%20%26%20-4%5C%5C3%20%26%20-12%20%26%20-15%5Cend%7Bmatrix%7D%5Cright%5D%20C%20%3D%20%5Cleft%5B%5Cbegin%7Bmatrix%7D15%20%26%207%5Cend%7Bmatrix%7D%5Cright%5D%20D%20%3D%20%5Cleft%5B%5Cbegin%7Bmatrix%7D7%20%26%208%20%26%20-7%5C%5C5%20%26%205%20%26%202%5C%5C-2%20%26%20-2%20%26%20-1%5Cend%7Bmatrix%7D%5Cright%5D%20E%20%3D%20%5Cleft%5B%5Cbegin%7Bmatrix%7D15%20%26%20-12%5C%5C3%20%26%20-18%5C%5C17%20%26%206%5Cend%7Bmatrix%7D%5Cright%5D%20%20" alt="LaTeX: A = \left[\begin{matrix}-3 & 0\\-12 & 5\\14 & 3\end{matrix}\right] B = \left[\begin{matrix}-4 & -13 & -4\\3 & -12 & -15\end{matrix}\right] C = \left[\begin{matrix}15 & 7\end{matrix}\right] D = \left[\begin{matrix}7 & 8 & -7\\5 & 5 & 2\\-2 & -2 & -1\end{matrix}\right] E = \left[\begin{matrix}15 & -12\\3 & -18\\17 & 6\end{matrix}\right] " data-equation-content=" A = \left[\begin{matrix}-3 & 0\\-12 & 5\\14 & 3\end{matrix}\right] B = \left[\begin{matrix}-4 & -13 & -4\\3 & -12 & -15\end{matrix}\right] C = \left[\begin{matrix}15 & 7\end{matrix}\right] D = \left[\begin{matrix}7 & 8 & -7\\5 & 5 & 2\\-2 & -2 & -1\end{matrix}\right] E = \left[\begin{matrix}15 & -12\\3 & -18\\17 & 6\end{matrix}\right] " />
<ol type="a">
<li>Find <img class="equation_image" title=" \displaystyle A + E " src="/equation_images/%20%5Cdisplaystyle%20A%20%2B%20E%20" alt="LaTeX: \displaystyle A + E " data-equation-content=" \displaystyle A + E " /> and <img class="equation_image" title=" \displaystyle C + D " src="/equation_images/%20%5Cdisplaystyle%20C%20%2B%20D%20" alt="LaTeX: \displaystyle C + D " data-equation-content=" \displaystyle C + D " /> </li>
<li>Find <img class="equation_image" title=" \displaystyle BC " src="/equation_images/%20%5Cdisplaystyle%20BC%20" alt="LaTeX: \displaystyle BC " data-equation-content=" \displaystyle BC " /> and <img class="equation_image" title=" \displaystyle CB " src="/equation_images/%20%5Cdisplaystyle%20CB%20" alt="LaTeX: \displaystyle CB " data-equation-content=" \displaystyle CB " /> </li>
<li>Find the inverse of Matrix D, that is <img class="equation_image" title=" \displaystyle D^{-1} " src="/equation_images/%20%5Cdisplaystyle%20D%5E%7B-1%7D%20" alt="LaTeX: \displaystyle D^{-1} " data-equation-content=" \displaystyle D^{-1} " /> </li>
</ol>
</p> </p><p> <p>
<ol type="a">
<li> <img class="equation_image" title=" \displaystyle \left[\begin{matrix}-3 & 0\\-12 & 5\\14 & 3\end{matrix}\right]+\left[\begin{matrix}15 & -12\\3 & -18\\17 & 6\end{matrix}\right]=\left[\begin{matrix}12 & -12\\-9 & -13\\31 & 9\end{matrix}\right] " src="/equation_images/%20%5Cdisplaystyle%20%5Cleft%5B%5Cbegin%7Bmatrix%7D-3%20%26%200%5C%5C-12%20%26%205%5C%5C14%20%26%203%5Cend%7Bmatrix%7D%5Cright%5D%2B%5Cleft%5B%5Cbegin%7Bmatrix%7D15%20%26%20-12%5C%5C3%20%26%20-18%5C%5C17%20%26%206%5Cend%7Bmatrix%7D%5Cright%5D%3D%5Cleft%5B%5Cbegin%7Bmatrix%7D12%20%26%20-12%5C%5C-9%20%26%20-13%5C%5C31%20%26%209%5Cend%7Bmatrix%7D%5Cright%5D%20" alt="LaTeX: \displaystyle \left[\begin{matrix}-3 & 0\\-12 & 5\\14 & 3\end{matrix}\right]+\left[\begin{matrix}15 & -12\\3 & -18\\17 & 6\end{matrix}\right]=\left[\begin{matrix}12 & -12\\-9 & -13\\31 & 9\end{matrix}\right] " data-equation-content=" \displaystyle \left[\begin{matrix}-3 & 0\\-12 & 5\\14 & 3\end{matrix}\right]+\left[\begin{matrix}15 & -12\\3 & -18\\17 & 6\end{matrix}\right]=\left[\begin{matrix}12 & -12\\-9 & -13\\31 & 9\end{matrix}\right] " /> The sum is undefined. The matricies do not have the same shape.</li>
<li> The product is undefined. <img class="equation_image" title=" \displaystyle \left[\begin{matrix}15 & 7\end{matrix}\right]\left[\begin{matrix}-4 & -13 & -4\\3 & -12 & -15\end{matrix}\right]=\left[\begin{matrix}-39 & -279 & -165\end{matrix}\right] " src="/equation_images/%20%5Cdisplaystyle%20%5Cleft%5B%5Cbegin%7Bmatrix%7D15%20%26%207%5Cend%7Bmatrix%7D%5Cright%5D%5Cleft%5B%5Cbegin%7Bmatrix%7D-4%20%26%20-13%20%26%20-4%5C%5C3%20%26%20-12%20%26%20-15%5Cend%7Bmatrix%7D%5Cright%5D%3D%5Cleft%5B%5Cbegin%7Bmatrix%7D-39%20%26%20-279%20%26%20-165%5Cend%7Bmatrix%7D%5Cright%5D%20" alt="LaTeX: \displaystyle \left[\begin{matrix}15 & 7\end{matrix}\right]\left[\begin{matrix}-4 & -13 & -4\\3 & -12 & -15\end{matrix}\right]=\left[\begin{matrix}-39 & -279 & -165\end{matrix}\right] " data-equation-content=" \displaystyle \left[\begin{matrix}15 & 7\end{matrix}\right]\left[\begin{matrix}-4 & -13 & -4\\3 & -12 & -15\end{matrix}\right]=\left[\begin{matrix}-39 & -279 & -165\end{matrix}\right] " /> </li>
<li> <img class="equation_image" title=" \displaystyle \left[\begin{matrix}-1 & 22 & 51\\1 & -21 & -49\\0 & -2 & -5\end{matrix}\right] " src="/equation_images/%20%5Cdisplaystyle%20%5Cleft%5B%5Cbegin%7Bmatrix%7D-1%20%26%2022%20%26%2051%5C%5C1%20%26%20-21%20%26%20-49%5C%5C0%20%26%20-2%20%26%20-5%5Cend%7Bmatrix%7D%5Cright%5D%20" alt="LaTeX: \displaystyle \left[\begin{matrix}-1 & 22 & 51\\1 & -21 & -49\\0 & -2 & -5\end{matrix}\right] " data-equation-content=" \displaystyle \left[\begin{matrix}-1 & 22 & 51\\1 & -21 & -49\\0 & -2 & -5\end{matrix}\right] " /> </li>
</ol></p> </p>