The matrix AB is:
[tex]AB=\left[\begin{array}{ccc}50&44&-43\\31&16&7\\37&119&-94\end{array}\right][/tex]
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This question is solved using multiplication of matrices.
To do this, we multiply the lines of the first matrix by the columns of the second.
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In this question:
The first matrix is:
[tex]A =\left[\begin{array}{ccc}5&7&2\\4&-1&3\\6&8&-5\end{array}\right][/tex]
The second matrix is:
[tex]B = \left[\begin{array}{ccc}6&11&-4\\2&1&-5\\3&-9&6\end{array}\right][/tex]
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First line:
- The first element is the multiplication of the first line of A by the first column of B. So:
- [tex]AB_{1,1} = 5\times6 + 7\times2 + 2\times3 = 50[/tex]
- The second element is the multiplication of the first line of A by the second column of B. So:
- [tex]AB_{1,2} = 5\times11 + 7\times1 + 2\times -9 = 44[/tex]
- The third element is the multiplication of the first line of A by the third column of B. So:
- [tex]AB_{1,3} = 5\times -4 + 7\times-5 + 2\times6 = -43[/tex]
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Second line:
Second line of A by each column of B, so:
[tex]A_{2,1} = 4\times6 - 1\times 2 + 3\times3 = 31[/tex]
[tex]A_{2,2} = 4\times11 - 1\times 1 + 3\times (-9) = 16[/tex]
[tex]A_{2,3} = 4\times(-4) - 1\times (-5) + 3\times 6 = 7[/tex]
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Third line:
Third line of A by each column of B, so:
[tex]A_{3,1} = 6\times6 + 8\times 2 - 5\times3 = 37[/tex]
[tex]A_{3,2} = 6\times11 + 8\times 1 - 5\times (-9) = 119[/tex]
[tex]A_{3,3} = 6\times(-4) + 8\times (-5) - 5\times 6 = -94[/tex]
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Thus, the matrix is:
[tex]AB=\left[\begin{array}{ccc}50&44&-43\\31&16&7\\37&119&-94\end{array}\right][/tex]
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