Notes

 

Practice Problems

\(\textbf{1)}\) What is the \(3\times3\) identity matrix?

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

 

\(\textbf{2)}\) What is the \(2\times2\) identity matrix?

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

 

\(\textbf{3)}\) What is the \(4\times4\) identity matrix?

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

 

\(\textbf{4)}\) Perform the following matrix multiplication
\(\left[ {\begin{array}{cc}5 & -4 \\2 & 3 \\ \end{array} } \right]\left[ {\begin{array}{cc}1 & 0 \\0 & 1 \\ \end{array} } \right]\)

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The answer is \(\left[ {\begin{array}{cc}5 & -4 \\2 & 3 \\ \end{array} } \right]\)

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\(\left[ {\begin{array}{cc}5 & -4 \\2 & 3 \\ \end{array} } \right]\left[ {\begin{array}{cc}1 & 0 \\0 & 1 \\ \end{array} } \right]=\left[ {\begin{array}{cc}(5)(1)+(-4)(0) & (5)(0)+(-4)(1) \\(2)(1)+(3)(0) & (2)(0)+(3)(1) \\ \end{array} } \right]\)
\(\left[ {\begin{array}{cc}5 & -4 \\2 & 3 \\ \end{array} } \right]\left[ {\begin{array}{cc}1 & 0 \\0 & 1 \\ \end{array} } \right]=\left[ {\begin{array}{cc}5 & -4 \\2 & 3 \\ \end{array} } \right]\)

 

\(\textbf{5)}\) Perform the following matrix multiplication
\(\left[ {\begin{array}{cc}9 & 8 \\7 & 5 \\ \end{array} } \right]\left[ {\begin{array}{cc}1 & 0 \\0 & 1 \\ \end{array} } \right]\)

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The answer is \(\left[ {\begin{array}{cc}9 & 8 \\7 & 5 \\ \end{array} } \right]\)

 

\(\textbf{6)}\) Perform the following matrix multiplication
\(\left[ {\begin{array}{ccc}1 & 2 & 3 \\4 & 5 & 6 \\7 & 8 & 9 \\ \end{array} } \right]\left[ {\begin{array}{ccc}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\ \end{array} } \right]\)

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The answer is \(\left[ {\begin{array}{ccc}1 & 2 & 3 \\4 & 5 & 6 \\7 & 8 & 9 \\ \end{array} } \right]\)

 

\(\textbf{7)}\) Perform the following matrix multiplication
\(\left[ {\begin{array}{ccc}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\ \end{array} } \right]\left[ {\begin{array}{ccc}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\ \end{array} } \right]\)

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

 

 

See Related Pages\(\)

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

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

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

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

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

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

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

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

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

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