17Calculus Precalculus - Matrix Multiplication
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As discussed on the basics of matrices page, multiplying two matrices together is not as simple as multiplying two numbers together. Here is a good introduction video and example. He explains quite thoroughly how to do matrix multiplication.
Khan Academy - Matrix multiplication introduction [6min-25secs]
video by Khan Academy |
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Let's work a few practice problems.
Unless otherwise instructed, perform matrix multiplication on these matrices.
Unless otherwise instructed, perform matrix multiplication on these matrices.
\(\left[\begin{array}{rr} 1 & 2 \\ 3 & 4 \end{array}\right] \left[\begin{array}{rr} 5 & 6 \\ 7 & 8 \end{array}\right] \)
Problem Statement |
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Unless otherwise instructed, perform matrix multiplication on these matrices.
\(\left[\begin{array}{rr} 1 & 2 \\ 3 & 4 \end{array}\right] \left[\begin{array}{rr} 5 & 6 \\ 7 & 8 \end{array}\right] \)
Final Answer |
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\(\left[\begin{array}{rr} 19 & 22 \\ 43 & 50 \end{array}\right]\)
Problem Statement |
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Unless otherwise instructed, perform matrix multiplication on these matrices.
\(\left[\begin{array}{rr} 1 & 2 \\ 3 & 4 \end{array}\right] \left[\begin{array}{rr} 5 & 6 \\ 7 & 8 \end{array}\right] \)
Solution |
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2015 video
video by PatrickJMT |
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Final Answer |
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\(\left[\begin{array}{rr} 19 & 22 \\ 43 & 50 \end{array}\right]\) |
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close solution
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Unless otherwise instructed, perform matrix multiplication on these matrices.
\(\displaystyle{ \left[\begin{array}{rr} 2 & 1 \\ 3 & 4 \\ 5 & 6 \end{array}\right] \left[\begin{array}{rrr} 1 & 3 & 6 \\ 2 & 4 & 5 \end{array}\right] }\)
Problem Statement |
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Unless otherwise instructed, perform matrix multiplication on these matrices.
\(\displaystyle{ \left[\begin{array}{rr} 2 & 1 \\ 3 & 4 \\ 5 & 6 \end{array}\right] \left[\begin{array}{rrr} 1 & 3 & 6 \\ 2 & 4 & 5 \end{array}\right] }\)
Final Answer |
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\(\displaystyle{\left[\begin{array}{rrr} 4 & 10 & 17 \\ 11 & 25 & 38 \\ 17 & 39 & 60\end{array}\right]}\)
Problem Statement |
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Unless otherwise instructed, perform matrix multiplication on these matrices.
\(\displaystyle{ \left[\begin{array}{rr} 2 & 1 \\ 3 & 4 \\ 5 & 6 \end{array}\right] \left[\begin{array}{rrr} 1 & 3 & 6 \\ 2 & 4 & 5 \end{array}\right] }\)
Solution |
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2016 video
video by PatrickJMT |
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Final Answer |
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\(\displaystyle{\left[\begin{array}{rrr} 4 & 10 & 17 \\ 11 & 25 & 38 \\ 17 & 39 & 60\end{array}\right]}\) |
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Unless otherwise instructed, perform matrix multiplication on these matrices.
\(\displaystyle{ \left[\begin{array}{rrr} 1 & 2 & 3 \\ 0 & -1 & 5\end{array}\right] \left[\begin{array}{r} 1 \\ 2 \\ -4\end{array}\right] }\)
Problem Statement |
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Unless otherwise instructed, perform matrix multiplication on these matrices.
\(\displaystyle{ \left[\begin{array}{rrr} 1 & 2 & 3 \\ 0 & -1 & 5\end{array}\right] \left[\begin{array}{r} 1 \\ 2 \\ -4\end{array}\right] }\)
Final Answer |
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\(\displaystyle{\left[\begin{array}{r}-7 \\ -22\end{array}\right]}\)
Problem Statement |
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Unless otherwise instructed, perform matrix multiplication on these matrices.
\(\displaystyle{ \left[\begin{array}{rrr} 1 & 2 & 3 \\ 0 & -1 & 5\end{array}\right] \left[\begin{array}{r} 1 \\ 2 \\ -4\end{array}\right] }\)
Solution |
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2018 video
video by PatrickJMT |
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Final Answer |
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\(\displaystyle{\left[\begin{array}{r}-7 \\ -22\end{array}\right]}\) |
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close solution
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Unless otherwise instructed, perform matrix multiplication on these matrices.
\(\displaystyle{\left[\begin{array}{rrr} 4 & 1 & 3 \\ 2 & -5 & 7\end{array}\right] \left[\begin{array}{rr} -2 & 3 \\ -4 & -2 \\ 5 & 1 \end{array}\right] }\)
Problem Statement |
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Unless otherwise instructed, perform matrix multiplication on these matrices.
\(\displaystyle{\left[\begin{array}{rrr} 4 & 1 & 3 \\ 2 & -5 & 7\end{array}\right] \left[\begin{array}{rr} -2 & 3 \\ -4 & -2 \\ 5 & 1 \end{array}\right] }\)
Final Answer |
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\(\displaystyle{\left[\begin{array}{rr} 3 & 13 \\ 51 & 23\end{array}\right]}\)
Problem Statement |
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Unless otherwise instructed, perform matrix multiplication on these matrices.
\(\displaystyle{\left[\begin{array}{rrr} 4 & 1 & 3 \\ 2 & -5 & 7\end{array}\right] \left[\begin{array}{rr} -2 & 3 \\ -4 & -2 \\ 5 & 1 \end{array}\right] }\)
Solution |
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2019 video
video by MIP4U |
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Final Answer |
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\(\displaystyle{\left[\begin{array}{rr} 3 & 13 \\ 51 & 23\end{array}\right]}\) |
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close solution
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Unless otherwise instructed, perform matrix multiplication on these matrices.
\(\displaystyle{\left[\begin{array}{rr} -2 & 3 \\ -4 & -2 \\ 5 & 1 \end{array}\right] \left[\begin{array}{rrr} 4 & 1 & 3 \\ 2 & -5 & 7\end{array}\right] }\)
Problem Statement |
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Unless otherwise instructed, perform matrix multiplication on these matrices.
\(\displaystyle{\left[\begin{array}{rr} -2 & 3 \\ -4 & -2 \\ 5 & 1 \end{array}\right] \left[\begin{array}{rrr} 4 & 1 & 3 \\ 2 & -5 & 7\end{array}\right] }\)
Final Answer |
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\(\displaystyle{\left[\begin{array}{rrr} -2 & -17 & 15 \\ -20 & 6 & -26 \\ 22 & 0 & 22\end{array}\right]}\)
Problem Statement |
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Unless otherwise instructed, perform matrix multiplication on these matrices.
\(\displaystyle{\left[\begin{array}{rr} -2 & 3 \\ -4 & -2 \\ 5 & 1 \end{array}\right] \left[\begin{array}{rrr} 4 & 1 & 3 \\ 2 & -5 & 7\end{array}\right] }\)
Solution |
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2020 video
video by MIP4U |
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Final Answer |
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\(\displaystyle{\left[\begin{array}{rrr} -2 & -17 & 15 \\ -20 & 6 & -26 \\ 22 & 0 & 22\end{array}\right]}\) |
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Unless otherwise instructed, perform matrix multiplication on these matrices.
\( A = \left[\begin{array}{rrr} 3 & 1 & 4 \end{array}\right] \)
\( B = \left[\begin{array}{rr} 4 & 3 \\ 2 & 5 \\ 6 & 8 \end{array}\right] \)
Problem Statement |
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Unless otherwise instructed, perform matrix multiplication on these matrices.
\( A = \left[\begin{array}{rrr} 3 & 1 & 4 \end{array}\right] \)
\( B = \left[\begin{array}{rr} 4 & 3 \\ 2 & 5 \\ 6 & 8 \end{array}\right] \)
Final Answer |
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\( A \cdot B = \left[\begin{array}{rr} 38 & 46 \end{array} \right] \)
\( B \cdot A \) does not make sense.
Problem Statement |
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Unless otherwise instructed, perform matrix multiplication on these matrices.
\( A = \left[\begin{array}{rrr} 3 & 1 & 4 \end{array}\right] \)
\( B = \left[\begin{array}{rr} 4 & 3 \\ 2 & 5 \\ 6 & 8 \end{array}\right] \)
Solution |
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2877 video
Final Answer |
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\( A \cdot B = \left[\begin{array}{rr} 38 & 46 \end{array} \right] \) |
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Unless otherwise instructed, calculate \( A \cdot B \) using these matrices.
\( A = \left[\begin{array}{rr} 3 & 4 \\ 7 & 2 \\ 5 & 9 \end{array}\right] \)
\( B = \left[\begin{array}{rrr} 3 & 1 & 5 \\ 6 & 9 & 7 \end{array}\right] \)
Problem Statement |
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Unless otherwise instructed, calculate \( A \cdot B \) using these matrices.
\( A = \left[\begin{array}{rr} 3 & 4 \\ 7 & 2 \\ 5 & 9 \end{array}\right] \)
\( B = \left[\begin{array}{rrr} 3 & 1 & 5 \\ 6 & 9 & 7 \end{array}\right] \)
Final Answer |
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\( A \cdot B = \left[\begin{array}{rrr} 33 & 39 & 43 \\ 33 & 25 & 49 \\ 69 & 86 & 88 \end{array}\right] \)
Problem Statement |
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Unless otherwise instructed, calculate \( A \cdot B \) using these matrices.
\( A = \left[\begin{array}{rr} 3 & 4 \\ 7 & 2 \\ 5 & 9 \end{array}\right] \)
\( B = \left[\begin{array}{rrr} 3 & 1 & 5 \\ 6 & 9 & 7 \end{array}\right] \)
Solution |
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2878 video
Final Answer |
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\( A \cdot B = \left[\begin{array}{rrr} 33 & 39 & 43 \\ 33 & 25 & 49 \\ 69 & 86 & 88 \end{array}\right] \) |
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Unless otherwise instructed, perform matrix multiplication on these matrices.
\( A = \left[ \begin{array}{rrr} 2 & 5 & 6 \end{array} \right] \)
\( B = \left[\begin{array}{r} 3 \\ 4 \\ -5 \end{array} \right]\)
Problem Statement |
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Unless otherwise instructed, perform matrix multiplication on these matrices.
\( A = \left[ \begin{array}{rrr} 2 & 5 & 6 \end{array} \right] \)
\( B = \left[\begin{array}{r} 3 \\ 4 \\ -5 \end{array} \right]\)
Final Answer |
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\( AB = [ -4 ] \)
\( BA = \left[\begin{array}{rrr} 6 & 15 & 18 \\ 8 & 20 & 24 \\ -10 & -25 & -30 \end{array} \right] \)
Problem Statement |
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Unless otherwise instructed, perform matrix multiplication on these matrices.
\( A = \left[ \begin{array}{rrr} 2 & 5 & 6 \end{array} \right] \)
\( B = \left[\begin{array}{r} 3 \\ 4 \\ -5 \end{array} \right]\)
Solution |
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2879 video
Final Answer |
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\( AB = [ -4 ] \) |
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Unless otherwise instructed, perform matrix multiplication on these matrices.
\( A = \left[\begin{array}{rrr} 1 & 4 & -2 \\ 3 & 5 & -6 \end{array}\right] \)
\( B = \left[\begin{array}{rrrr} 5 & 2 & 8 & -1 \\ 3 & 6 & 4 & 5 \\ -2 & 9 & 7 & -3 \end{array}\right] \)
Problem Statement |
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Unless otherwise instructed, perform matrix multiplication on these matrices.
\( A = \left[\begin{array}{rrr} 1 & 4 & -2 \\ 3 & 5 & -6 \end{array}\right] \)
\( B = \left[\begin{array}{rrrr} 5 & 2 & 8 & -1 \\ 3 & 6 & 4 & 5 \\ -2 & 9 & 7 & -3 \end{array}\right] \)
Final Answer |
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\( AB = \left[\begin{array}{rrrr} 21 & 8 & 10 & 25 \\ 42 & -18 & 2 & 40 \end{array} \right] \)
Problem Statement |
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Unless otherwise instructed, perform matrix multiplication on these matrices.
\( A = \left[\begin{array}{rrr} 1 & 4 & -2 \\ 3 & 5 & -6 \end{array}\right] \)
\( B = \left[\begin{array}{rrrr} 5 & 2 & 8 & -1 \\ 3 & 6 & 4 & 5 \\ -2 & 9 & 7 & -3 \end{array}\right] \)
Solution |
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2880 video
Final Answer |
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\( AB = \left[\begin{array}{rrrr} 21 & 8 & 10 & 25 \\ 42 & -18 & 2 & 40 \end{array} \right] \) |
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Unless otherwise instructed, perform matrix multiplication on these matrices.
\( \left[\begin{array}{rrr} 2 & 1 & 3 \\ 0 & -1 & 4 \end{array}\right] \) \( \left[\begin{array}{r} 1 \\ 0 \\ 5 \end{array}\right] \)
Problem Statement |
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Unless otherwise instructed, perform matrix multiplication on these matrices.
\( \left[\begin{array}{rrr} 2 & 1 & 3 \\ 0 & -1 & 4 \end{array}\right] \) \( \left[\begin{array}{r} 1 \\ 0 \\ 5 \end{array}\right] \)
Final Answer |
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\( \left[\begin{array}{r} 17 \\ 20 \end{array}\right] \)
Problem Statement |
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Unless otherwise instructed, perform matrix multiplication on these matrices.
\( \left[\begin{array}{rrr} 2 & 1 & 3 \\ 0 & -1 & 4 \end{array}\right] \) \( \left[\begin{array}{r} 1 \\ 0 \\ 5 \end{array}\right] \)
Solution |
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2881 video
Final Answer |
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\( \left[\begin{array}{r} 17 \\ 20 \end{array}\right] \) |
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close solution
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Unless otherwise instructed, perform matrix multiplication on these matrices.
\( \left[\begin{array}{rr} 1 & 7 \\ -2 & 0 \end{array}\right] \) \( \left[\begin{array}{rr} 5 & 1 \\ 3 & 2 \end{array}\right] \)
Problem Statement |
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Unless otherwise instructed, perform matrix multiplication on these matrices.
\( \left[\begin{array}{rr} 1 & 7 \\ -2 & 0 \end{array}\right] \) \( \left[\begin{array}{rr} 5 & 1 \\ 3 & 2 \end{array}\right] \)
Final Answer |
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\( \left[\begin{array}{rr} 26 & 15 \\ -10 & -2 \end{array}\right] \)
Problem Statement |
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Unless otherwise instructed, perform matrix multiplication on these matrices.
\( \left[\begin{array}{rr} 1 & 7 \\ -2 & 0 \end{array}\right] \) \( \left[\begin{array}{rr} 5 & 1 \\ 3 & 2 \end{array}\right] \)
Solution |
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2882 video
Final Answer |
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\( \left[\begin{array}{rr} 26 & 15 \\ -10 & -2 \end{array}\right] \) |
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close solution
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Unless otherwise instructed, perform matrix multiplication on these matrices.
\( \left[\begin{array}{rr} 1 & 2 \end{array}\right] \) \( \left[\begin{array}{rr} 3 & -1 \\ 0 & 2 \end{array}\right] \)
Problem Statement |
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Unless otherwise instructed, perform matrix multiplication on these matrices.
\( \left[\begin{array}{rr} 1 & 2 \end{array}\right] \) \( \left[\begin{array}{rr} 3 & -1 \\ 0 & 2 \end{array}\right] \)
Final Answer |
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\( \left[\begin{array}{rr} 3 & 3 \end{array}\right] \)
Problem Statement |
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Unless otherwise instructed, perform matrix multiplication on these matrices.
\( \left[\begin{array}{rr} 1 & 2 \end{array}\right] \) \( \left[\begin{array}{rr} 3 & -1 \\ 0 & 2 \end{array}\right] \)
Solution |
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2883 video
Final Answer |
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\( \left[\begin{array}{rr} 3 & 3 \end{array}\right] \) |
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Unless otherwise instructed, perform matrix multiplication on these matrices.
\( \left[\begin{array}{rrr} 2 & 1 & 0 \\ 3 & 0 & -1 \end{array}\right] \) \( \left[\begin{array}{rr} 1 & 4 \\ 0 & 2 \\ 5 & 1 \end{array}\right] \)
Problem Statement |
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Unless otherwise instructed, perform matrix multiplication on these matrices.
\( \left[\begin{array}{rrr} 2 & 1 & 0 \\ 3 & 0 & -1 \end{array}\right] \) \( \left[\begin{array}{rr} 1 & 4 \\ 0 & 2 \\ 5 & 1 \end{array}\right] \)
Final Answer |
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\( \left[\begin{array}{rr} 2 & 10 \\ -2 & 11 \end{array}\right] \)
Problem Statement |
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Unless otherwise instructed, perform matrix multiplication on these matrices.
\( \left[\begin{array}{rrr} 2 & 1 & 0 \\ 3 & 0 & -1 \end{array}\right] \) \( \left[\begin{array}{rr} 1 & 4 \\ 0 & 2 \\ 5 & 1 \end{array}\right] \)
Solution |
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2884 video
Final Answer |
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\( \left[\begin{array}{rr} 2 & 10 \\ -2 & 11 \end{array}\right] \) |
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Unless otherwise instructed, perform matrix multiplication on these matrices.
\( \left[\begin{array}{rrr} -2 & 3 & 4 \\ 4 & 0 & 1 \end{array}\right] \) \( \left[\begin{array}{rrrr} 0 & -3 & 4 & -1 \\ 1 & 3 & 0 & -5 \\ -4 & 4 & 3 & -2 \end{array}\right] \)
Problem Statement |
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Unless otherwise instructed, perform matrix multiplication on these matrices.
\( \left[\begin{array}{rrr} -2 & 3 & 4 \\ 4 & 0 & 1 \end{array}\right] \) \( \left[\begin{array}{rrrr} 0 & -3 & 4 & -1 \\ 1 & 3 & 0 & -5 \\ -4 & 4 & 3 & -2 \end{array}\right] \)
Final Answer |
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\( \left[\begin{array}{rrrr} 19 & -1 & -20 & -5 \\ 4 & -16 & 13 & -2 \end{array}\right] \)
Problem Statement |
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Unless otherwise instructed, perform matrix multiplication on these matrices.
\( \left[\begin{array}{rrr} -2 & 3 & 4 \\ 4 & 0 & 1 \end{array}\right] \) \( \left[\begin{array}{rrrr} 0 & -3 & 4 & -1 \\ 1 & 3 & 0 & -5 \\ -4 & 4 & 3 & -2 \end{array}\right] \)
Solution |
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2885 video
Final Answer |
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\( \left[\begin{array}{rrrr} 19 & -1 & -20 & -5 \\ 4 & -16 & 13 & -2 \end{array}\right] \) |
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matrix multiplication 17calculus youtube playlist
Really UNDERSTAND Precalculus
Topics You Need To Understand For This Page
Trig Formulas
The Unit Circle
The Unit Circle [wikipedia]
Basic Trig Identities
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Set 1 - basic identities | |||
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\(\displaystyle{ \tan(t) = \frac{\sin(t)}{\cos(t)} }\) |
\(\displaystyle{ \cot(t) = \frac{\cos(t)}{\sin(t)} }\) |
\(\displaystyle{ \sec(t) = \frac{1}{\cos(t)} }\) |
\(\displaystyle{ \csc(t) = \frac{1}{\sin(t)} }\) |
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Set 2 - squared identities | ||
|---|---|---|
\( \sin^2t + \cos^2t = 1\) |
\( 1 + \tan^2t = \sec^2t\) |
\( 1 + \cot^2t = \csc^2t\) |
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Set 3 - double-angle formulas | |
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\( \sin(2t) = 2\sin(t)\cos(t)\) |
\(\displaystyle{ \cos(2t) = \cos^2(t) - \sin^2(t) }\) |
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Set 4 - half-angle formulas | |
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\(\displaystyle{ \sin^2(t) = \frac{1-\cos(2t)}{2} }\) |
\(\displaystyle{ \cos^2(t) = \frac{1+\cos(2t)}{2} }\) |
Trig Derivatives
\(\displaystyle{ \frac{d[\sin(t)]}{dt} = \cos(t) }\) |
\(\displaystyle{ \frac{d[\cos(t)]}{dt} = -\sin(t) }\) | |
\(\displaystyle{ \frac{d[\tan(t)]}{dt} = \sec^2(t) }\) |
\(\displaystyle{ \frac{d[\cot(t)]}{dt} = -\csc^2(t) }\) | |
\(\displaystyle{ \frac{d[\sec(t)]}{dt} = \sec(t)\tan(t) }\) |
\(\displaystyle{ \frac{d[\csc(t)]}{dt} = -\csc(t)\cot(t) }\) |
Inverse Trig Derivatives
\(\displaystyle{ \frac{d[\arcsin(t)]}{dt} = \frac{1}{\sqrt{1-t^2}} }\) |
\(\displaystyle{ \frac{d[\arccos(t)]}{dt} = -\frac{1}{\sqrt{1-t^2}} }\) | |
\(\displaystyle{ \frac{d[\arctan(t)]}{dt} = \frac{1}{1+t^2} }\) |
\(\displaystyle{ \frac{d[\arccot(t)]}{dt} = -\frac{1}{1+t^2} }\) | |
\(\displaystyle{ \frac{d[\arcsec(t)]}{dt} = \frac{1}{\abs{t}\sqrt{t^2 -1}} }\) |
\(\displaystyle{ \frac{d[\arccsc(t)]}{dt} = -\frac{1}{\abs{t}\sqrt{t^2 -1}} }\) |
Trig Integrals
\(\int{\sin(x)~dx} = -\cos(x)+C\) |
\(\int{\cos(x)~dx} = \sin(x)+C\) | |
\(\int{\tan(x)~dx} = -\ln\abs{\cos(x)}+C\) |
\(\int{\cot(x)~dx} = \ln\abs{\sin(x)}+C\) | |
\(\int{\sec(x)~dx} = \) \( \ln\abs{\sec(x)+\tan(x)}+C\) |
\(\int{\csc(x)~dx} = \) \( -\ln\abs{\csc(x)+\cot(x)}+C\) |
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Single Variable Calculus |
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Multi-Variable Calculus |
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Differential Equations |
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Precalculus |
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