Notes

 

Questions

\(\textbf{1)}\) Find the determinant. \(\left|{\begin{array}{cc} 3 & 1 \\ -5 & 4 \\ \end{array} } \right|\)

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The determinant is \(17\)

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\(\,\,\,\,\,\,\left|{\begin{array}{cc} a & b \\ c & d \\ \end{array} } \right|=(a)(d)-(b)(c)\)
\(\,\,\,\,\,\,\left|{\begin{array}{cc} 3 & 1 \\ -5 & 4 \\ \end{array} } \right|=(3)(4)-(1)(-5)\)
\(\,\,\,\,\,\,12+5=17\)

 

\(\textbf{2)}\) Find the determinant. \(\left|{\begin{array}{cc} 1 & 2 \\ 3 & 6 \\ \end{array} } \right|\)

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The determinant is \(0\)

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\(\,\,\,\,\,\,\left|{\begin{array}{cc} a & b \\ c & d \\ \end{array} } \right|=(a)(d)-(b)(c)\)
\(\,\,\,\,\,\,\left|{\begin{array}{cc} 1 & 2 \\ 3 & 6 \\ \end{array} } \right|=(1)(6)-(2)(3)\)
\(\,\,\,\,\,\,6-6=0\)

 

\(\textbf{3)}\) Find the determinant. \(\left|{\begin{array}{cc} 5 & -3 \\ 6 & 6 \\ \end{array} } \right|\)

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The determinant is \(48\)

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\(\,\,\,\,\,\,\left|{\begin{array}{cc} a & b \\ c & d \\ \end{array} } \right|=(a)(d)-(b)(c)\)
\(\,\,\,\,\,\,\left|{\begin{array}{cc} 5 & -3 \\ 6 & 6 \\ \end{array} } \right|=(5)(6)-(-3)(6)\)
\(\,\,\,\,\,\,30+18=48\)

 

\(\textbf{4)}\) Find the determinant. \(\left|{\begin{array}{cc} -1 & 4 \\ 4 & 9 \\ \end{array} } \right|\)

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The determinant is \(-25\)

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\(\,\,\,\,\,\,\left|{\begin{array}{cc} a & b \\ c & d \\ \end{array} } \right|=(a)(d)-(b)(c)\)
\(\,\,\,\,\,\,\left|{\begin{array}{cc} -1 & 4 \\ 4 & 9 \\ \end{array} } \right|=(-1)(9)-(4)(4)\)
\(\,\,\,\,\,\,-9-16=-25\)

 

\(\textbf{5)}\) Find the determinant. \(\left|{\begin{array}{cc} 2 & 4 \\ -6 & 8 \\ \end{array} } \right|\)

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The determinant is \(40\)

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\(\,\,\,\,\,\,\left|{\begin{array}{cc} a & b \\ c & d \\ \end{array} } \right|=(a)(d)-(b)(c)\)
\(\,\,\,\,\,\,\left|{\begin{array}{cc} 2 & 4 \\ -6 & 8 \\ \end{array} } \right|=(2)(8)-(4)(-6)\)
\(\,\,\,\,\,\,16+24=40\)

 

\(\textbf{6)}\) Find the determinant. \(\left|{\begin{array}{ccc} 1 & 2 & 3\\ 5 & 3 & -2 \\ 7 & 1 & 9 \\ \end{array} } \right|\)

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The determinant is \(-137\)

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\(\,\,\,\,\,\,\left|{\begin{array}{ccc} a & b & c\\ d & e & f \\ g & h & i \\ \end{array} } \right|\,\,\,=\,\,\,a \left| {\begin{array}{cc} e & f \\ h & i \\\end{array} } \right| -b \left| {\begin{array}{cc} d & f \\ g & i \\\end{array} } \right| +c \left| {\begin{array}{cc} d & e \\ g & h \\\end{array} } \right| \)

\(\,\,\,\,\,\,\left|{\begin{array}{ccc} 1 & 2 & 3\\ 5 & 3 & -2 \\ 7 & 1 & 9 \\ \end{array} } \right|\,\,\,=\,\,\,1 \left| {\begin{array}{cc} 3 & -2 \\ 1 & 9 \\\end{array} } \right| -2 \left| {\begin{array}{cc} 5 & -2 \\ 7 & 9 \\\end{array} } \right| +3 \left| {\begin{array}{cc} 5 & 3 \\ 7 & 1 \\\end{array} } \right| \) \(\,\,\,\,\,\,1\left[(3)(9)-(-2)(1)\right] \,\,-\,\,2\left[(5)(9)-(-2)(7)\right] \,\,+\,\,3\left[(5)(1)-(3)(7)\right]\) \(\,\,\,\,\,\,1\cdot\left[27+2\right] \,\,-\,\,2\cdot\left[45+14\right] \,\,+\,\,3\cdot\left[5-21\right]\)

\(\,\,\,\,\,\,29 -118 -48=-137\)

 

\(\textbf{7)}\) Find the determinant. \(\left|{\begin{array}{ccc} 1 & 2 & 3\\ 4 & 5 & 6 \\ 7 & 8 & 9 \\ \end{array} } \right|\)

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The determinant is \(0\)

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\(\,\,\,\,\,\,\left|{\begin{array}{ccc} a & b & c\\ d & e & f \\ g & h & i \\ \end{array} } \right|\,\,\,=\,\,\,a \left| {\begin{array}{cc} e & f \\ h & i \\\end{array} } \right| -b \left| {\begin{array}{cc} d & f \\ g & i \\\end{array} } \right| +c \left| {\begin{array}{cc} d & e \\ g & h \\\end{array} } \right| \)

\(\,\,\,\,\,\,\left|{\begin{array}{ccc} 1 & 2 & 3\\ 4 & 5 & 6 \\ 7 & 8 & 9 \\ \end{array} } \right|\,\,\,=\,\,\,1 \left| {\begin{array}{cc} 5 & 6 \\ 8 & 9 \\\end{array} } \right| -2 \left| {\begin{array}{cc} 4 & 6 \\ 7 & 9 \\\end{array} } \right| +3 \left| {\begin{array}{cc} 4 & 5 \\ 7 & 8 \\\end{array} } \right| \) \(\,\,\,\,\,\,1\left[(5)(9)-(6)(8)\right] \,\,-\,\,2\left[(4)(9)-(6)(7)\right] \,\,+\,\,3\left[(4)(8)-(5)(7)\right]\) \(\,\,\,\,\,\,1\cdot\left[45-48\right] \,\,-\,\,2\cdot\left[36-42\right] \,\,+\,\,3\cdot\left[32-35\right]\)

\(\,\,\,\,\,\,-3 +12 -9=0\)

 

 

See Related Pages\(\)

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

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

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

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

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

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

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

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

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

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

 

External Links

\( \text{ 3×3 Matrix Determinant Calculator (Wolfram Alpha)}\)