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Calculate the product $\\mathbf{AB}$

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First set up the size of the answer matrix (choose the correct number of rows and columns in the boxes) and then input the entries.

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$\\mathbf{AB}$ =   [[0]]

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Calculate the product $\\mathbf{BA}$

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First set up the size of the answer matrix (choose the correct number of rows and columns in the boxes) and then input the entries.

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$\\mathbf{BA}$ = [[0]]

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\$$\\mathbf{A}\\mathbf{B}=\\begin{pmatrix} \\var{a11}&\\var{a12}&\\var{a13}\\\\ \\var{a21}&\\var{a22}&\\var{a23}\\\\ \\end{pmatrix}\\begin{pmatrix} \\var{b11}&\\var{b12}\\\\ \\var{b21}&\\var{b22}\\\\ \\var{b31}&\\var{b32}\\\\\\end{pmatrix}\$$

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Remember multiplication of matrices is carried out by multiplying the rows of the first matrix, \$$\\mathbf{A}\$$, by the columns of the second matrix, \$$\\mathbf{B}\$$.

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\$$\\begin{pmatrix} \\var{a11}&\\var{a12}&\\var{a13}\\\\ \\var{a21}&\\var{a22}&\\var{a23} \\end{pmatrix}\\begin{pmatrix} \\var{b11}&\\var{b12}\\\\ \\var{b21}&\\var{b22}\\\\ \\var{b31}&\\var{b32}\\end{pmatrix}=\\begin{pmatrix}\\var{a11}*\\var{b11}+\\var{a12}*\\var{b21}+\\var{a13}*\\var{b31}&\\var{a11}*\\var{b12}+\\var{a12}*\\var{b22}+\\var{a13}*\\var{b32}\\\\ \\var{a21}*\\var{b11}+\\var{a22}*\\var{b21}+\\var{a23}*\\var{b31}&\\var{a21}*\\var{b12}+\\var{a22}*\\var{b22}+\\var{a23}*\\var{b32}\\end{pmatrix}\$$

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\$$\\begin{pmatrix} \\var{a11}&\\var{a12}&\\var{a13}\\\\ \\var{a21}&\\var{a22}&\\var{a23} \\end{pmatrix}\\begin{pmatrix} \\var{b11}&\\var{b12}\\\\ \\var{b21}&\\var{b22}\\\\ \\var{b31}&\\var{b32}\\end{pmatrix}=\\begin{pmatrix}\\simplify{{a11}*{b11}+{a12}*{b21}+{a13}*{b31}}&\\simplify{{a11}*{b12}+{a12}*{b22}+{a13}*{b32}}\\\\ \\simplify{{a21}*{b11}+{a22}*{b21}+{a23}*{b31}}&\\simplify{{a21}*{b12}+{a22}*{b22}+{a23}*{b32}}\\end{pmatrix}\$$

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To evaluate \$$\\mathbf{B}\\mathbf{A}\$$ we swap their positions and this time multiply the rows of \$$\\mathbf{B}\$$ by the columns of \$$\\mathbf{A}\$$

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\$$\\mathbf{B}\\mathbf{A}=\\begin{pmatrix} \\var{b11}&\\var{b12}\\\\ \\var{b21}&\\var{b22}\\\\ \\var{b31}&\\var{b32}\\end{pmatrix}\\begin{pmatrix} \\var{a11}&\\var{a12}&\\var{a13}\\\\ \\var{a21}&\\var{a22}&\\var{a23}\\\\ \\end{pmatrix}=\\begin{pmatrix} \\var{ba11}&\\var{ba12}&\\var{ba13}\\\\ \\var{ba21}&\\var{ba22}&\\var{ba23}\\\\ \\var{ba31}&\\var{ba32}&\\var{ba33}\\end{pmatrix}\$$

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Multiplication of two matrices.

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Given two matrices:

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\$$\\mathbf{A}=\\begin{pmatrix} \\var{a11}&\\var{a12}&\\var{a13}\\\\ \\var{a21}&\\var{a22}&\\var{a23}\\\\ \\end{pmatrix}\\;\\;and \\;\\;\\mathbf{B}=\\begin{pmatrix} \\var{b11}&\\var{b12}\\\\ \\var{b21}&\\var{b22}\\\\ \\var{b31}&\\var{b32}\\\\\\end{pmatrix}\$$

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", "type": "question", "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}, {"name": "John Steele", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2218/"}]}]}], "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}, {"name": "John Steele", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2218/"}]}