Notes

 

Questions

\(\textbf{1)}\) Find the inverse. \(\left[ {\begin{array}{cc}
5 & 2 \\
3 & 1 \\
\end{array} } \right]\)

Show Answer

The inverse is \(\left[ {\begin{array}{cc}
-1 & 2 \\
3 & -5 \\
\end{array} } \right]\)

Show Work

\(\textbf{Notes}\)
\(\,\,\,\,\,\,A=\left[ {\begin{array}{cc}
a & b \\
c & d \\
\end{array} } \right] \Rightarrow A^{-1}=\displaystyle \frac{1}{(a)(d)-(b)(c)}\left[ {\begin{array}{cc}
d & -b \\
-c & a \\
\end{array} } \right]\)
\(\textbf{Plug in given matrix}\)
\(\,\,\,\,\,\,A=\left[ {\begin{array}{cc}
5 & 2 \\
3 & 1 \\
\end{array} } \right] \Rightarrow A^{-1}=\displaystyle \frac{1}{(5)(1)-(2)(3)}\left[ {\begin{array}{cc}
1 & -2 \\
-3 & 5 \\
\end{array} } \right]\)
\(\textbf{Simplify}\)
\(\,\,\,\,\,\,\displaystyle \frac{1}{-1}\left[ {\begin{array}{cc}
1 & -2 \\
-3 & 5 \\
\end{array} } \right]\)
\(\,\,\,\,\,\,\left[ {\begin{array}{cc}
-1 & 2 \\
3 & -5 \\
\end{array} } \right]\)

 

\(\textbf{2)}\) Find the inverse. \(\left[ {\begin{array}{cc}
2 & -3 \\
-4 & 6 \\
\end{array} } \right]\)

Show Answer

No inverse exists

 

\(\textbf{3)}\) Find the inverse. \(\left[ {\begin{array}{cc}
3 & 4 \\
-5 & 2 \\
\end{array} } \right]\)

Show Answer

The inverse is \(\left[ {\begin{array}{cc}
\displaystyle\frac{1}{13} & \displaystyle\frac{-2}{13} \\
\displaystyle\frac{5}{26} & \displaystyle\frac{3}{26} \\
\end{array} } \right]\)

 

\(\textbf{4)}\) Find the inverse. \(\left[ {\begin{array}{ccc}
5 & -3 & 4 \\
7 & 1 & 2 \\
\end{array} } \right]\)

Show Answer

No inverse exists

 

\(\textbf{5)}\) Multiply these matrices to verify they are inverses.\(\left[ {\begin{array}{cc}
1 & 1\\
4 & 5 \\
\end{array} } \right]\left[ {\begin{array}{cc}
5 & -1 \\
-4 & 1 \\
\end{array} } \right]\)
See Video for verification steps

 

\(\textbf{6)}\) What is the 3X3 identity matrix?

Show Answer

The answer is \(\left[ {\begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 \\
\end{array} } \right]\)

 

\(\textbf{7)}\) If \(AX=B \text{ where } A = \left[ {\begin{array}{ccc}
1 & 1 & 2 \\
3 & 4 & 5 \\
0 & 1 & 0 \\
\end{array} } \right] \text{ and } B=\left[ {\begin{array}{cc}
1 & 3 \\
1 & 1 \\
1 & 2 \\
\end{array} } \right], \text{ what is } X\)?

Show Answer

The answer is \(X = \left[ {\begin{array}{ccc}
-6 & -19 \\
1 & 2 \\
3 & 10 \\
\end{array} } \right]\)

 

 

See Related Pages\(\)

\(\bullet\text{ Intro to Matrices}\)
\(\,\,\,\,\,\,\,\,\)\(\left[ {\begin{array}{ccc}4 & -5 & 2 \\1 & 0 & 3 \\\end{array} } \right]\)

\(\bullet\text{ Matrix Operations}\)
\(\,\,\,\,\,\,\,\,\)\( \left[ {\begin{array}{ccc}3 & 45 & 6 \\-8 & 2 & 4 \\1 & 0 & 3 \\\end{array} } \right]\)\(+\left[ {\begin{array}{ccc}3 & 45 & 6 \\-8 & 2 & 4 \\1 & 0 & 3 \\\end{array} } \right]\)

\(\bullet\text{ Multiplying Matrices}\)
\(\,\,\,\,\,\,\,\,\)\(\left[{\begin{array}{cc} 1 & 2 \\ -3 & -4 \\\end{array} } \right]\)\(\left[ {\begin{array}{cc}6 & -3 \\5 & 0 \\\end{array} } \right]\)

\(\bullet\text{ Determinants}\)
\(\,\,\,\,\,\,\,\,\)\(\left|{\begin{array}{cc} a & b \\ c & d \\ \end{array} } \right|=ad-bc\)

\(\bullet\text{ Cramer’s Rule}\)
\(\,\,\,\,\,\,\,\,\text{ax+by=e } \& \text{ cx+dy=f}…\)

\(\bullet\text{ Identity Matrix}\)
\(\,\,\,\,\,\,\,\,\)\(\left[{\begin{array}{ccc} 1 & 0 & 0\\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} } \right]\)

\(\bullet\text{ Identity and Inverse Matrices}\)
\(\,\,\,\,\,\,\,\,A^{-1}=\displaystyle\frac{1}{ad-bc}\left[{\begin{array}{cc} a & b \\ c & d \\ \end{array} } \right]\)

\(\bullet\text{ Transpose Matrix}\)
\(\,\,\,\,\,\,\,\,\left[{\begin{array}{ccc} 1 \\ 2 \\ 5 \\ \end{array} } \right]\Rightarrow\left[{\begin{array}{c} 1 & 2 & 5 \end{array} } \right]\)

\(\bullet\text{ Rotation Matrix}\)
\(\,\,\,\,\,\,\,\,\)\(R(\theta)=\left[{\begin{array}{cc}\cos{\theta} & -\sin{\theta} \\\sin{\theta} & \cos{\theta} \\\end{array} } \right]\)

\(\bullet\text{ Eigenvectors and Eigenvalues}\)
\(\,\,\,\,\,\,\,\,(A-\lambda I)\vec{v}=\vec{0}\)