Multiplication of $2 \\times 2$ matrices.
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\\mathbf{AB} &= \\var{A}\\var{B} \\\\
&= \\begin{pmatrix} \\simplify[]{ {a[0][0]}*{b[0][0]}+{a[0][1]}*{b[1][0]} } & \\simplify[]{ {a[0][0]}*{b[0][1]} + {a[0][1]}*{b[1][1]} } \\\\ \\simplify[]{ {a[1][0]}*{b[0][0]} + {a[1][1]}*{b[1][0]} } & \\simplify[]{ {a[1][0]}*{b[0][1]} + {a[1][1]}*{b[1][1]} } \\end{pmatrix} \\\\
&= \\var{a*b}
\\end{align}
\\begin{align}
\\mathbf{BA} &= \\var{B}\\var{A} \\\\
&= \\begin{pmatrix} \\simplify[]{ {b[0][0]}*{a[0][0]}+{b[0][1]}*{a[1][0]} } & \\simplify[]{ {b[0][0]}*{a[0][1]} + {b[0][1]}*{a[1][1]} } \\\\ \\simplify[]{ {b[1][0]}*{a[0][0]} + {b[1][1]}*{a[1][0]} } & \\simplify[]{ {b[1][0]}*{a[0][1]} + {b[1][1]}*{a[1][1]} } \\end{pmatrix} \\\\
&= \\var{b*a}
\\end{align}
\\begin{align}
\\mathbf{CB} &= \\var{C}\\var{B} \\\\
&= \\begin{pmatrix} \\simplify[]{ {c[0][0]}*{b[0][0]}+{c[0][1]}*{b[1][0]} } & \\simplify[]{ {c[0][0]}*{b[0][1]} + {c[0][1]}*{b[1][1]} } \\\\ \\simplify[]{ {c[1][0]}*{b[0][0]} + {c[1][1]}*{b[1][0]} } & \\simplify[]{ {c[1][0]}*{b[0][1]} + {c[1][1]}*{b[1][1]} } \\end{pmatrix} \\\\
&= \\var{c*b}
\\end{align}
\\begin{align}
\\mathbf{AC} &= \\var{A}\\var{C} \\\\
&= \\begin{pmatrix} \\simplify[]{ {a[0][0]}*{c[0][0]}+{a[0][1]}*{c[1][0]} } & \\simplify[]{ {a[0][0]}*{c[0][1]} + {a[0][1]}*{c[1][1]} } \\\\ \\simplify[]{ {a[1][0]}*{c[0][0]} + {a[1][1]}*{c[1][0]} } & \\simplify[]{ {a[1][0]}*{c[0][1]} + {a[1][1]}*{c[1][1]} } \\end{pmatrix} \\\\
&= \\var{a*c}
\\end{align}
$\\mathbf{AB} = \\var{A}\\var{B} = $ [[0]]
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", "variableReplacementStrategy": "originalfirst"}], "statement": "Let
\n\\begin{align} \\mathbf{A} &= \\var{a}, & \\mathbf{B} &= \\var{b}, & \\mathbf{C} &= \\var{c} \\end{align}
\nCalculate the following products of these matrices:
", "type": "question", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Timur Zaripov", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3272/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Timur Zaripov", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3272/"}]}