Addition and subtraction of matrices; multiplication by scalar.
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\n\\[A=\\var{A}\\qquad B=\\var{B}\\qquad C=\\var{C}\\]
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", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "The matrices $A$, $B$, $C$ and $D$ are defined as:
\n\\[A=\\var{A}\\qquad B=\\var{B}\\qquad C=\\var{C}\\qquad D=\\var{D}\\]
\nMultiply the matrices, as stated below. You should adjust the number of rows and columns to obtain the correct shape of the answer matrix.
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\n\\[A=\\var{M}\\]
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\n\\[\\begin{array}[ccc]++&-&+\\\\-&+&-\\\\+&-&+\\end{array}\\]
\n", "type": "matrix", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "allowResize": false, "numColumns": "3", "marks": "9", "correctAnswer": "matrix([a11,-a12,a13],[-a21,a22,-a23],[a31,-a32,a33])", "tolerance": 0, "markPerCell": true, "variableReplacements": [], "allowFractions": false}, {"showFeedbackIcon": true, "scripts": {}, "numRows": "3", "correctAnswerFractions": false, "prompt": "Write down the adjoint of $A$. (Transpose of the cofactor matrix.)
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\n| 1 | \n[[1]] | \n
| [[0]] | \n
The matrix $A$ is:
\n\\[A=\\var{M}\\]
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"}, "variablesTest": {"maxRuns": 100, "condition": "det(M)<>0"}, "type": "question"}, {"name": "Solve system of three simultaneous equations with matrices", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Martin Jones", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/145/"}], "tags": [], "metadata": {"description": "Use matrices to solve a system of three simultaneous equations
", "licence": "None specified"}, "statement": "You are going to solve the following system of simultaneous equations:
\n\\[\\begin{eqnarray}\\simplify{{m11}*x+{m12}*y+{m13}*z}&=&\\var{r1}\\\\
\\simplify{{m21}*x+{m22}*y+{m23}*z}&=&\\var{r2}\\\\
\\simplify{{m31}*x+{m32}*y+{m33}*z}&=&\\var{r3}\\\\\\end{eqnarray}\\]
The system of equations can be written in the form
\n\\[M\\Bigg(\\begin{matrix}x\\\\y\\\\z\\end{matrix}\\Bigg)=\\Bigg(\\begin{matrix}\\var{r1}\\\\\\var{r2}\\\\\\var{r3}\\end{matrix}\\Bigg)\\]
\nWrite down the matrix $M$.
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\n| 1 | \n[[1]] | \n
| [[0]] | \n
By multiplying on the left by $M^{-1}$ we obtain
\n\\[\\Bigg(\\begin{matrix}x\\\\y\\\\z\\end{matrix}\\Bigg)=M^{-1}\\Bigg(\\begin{matrix}\\var{r1}\\\\\\var{r2}\\\\\\var{r3}\\end{matrix}\\Bigg)\\]
\nCalculate the values of $x$, $y$ and $z$.
\nFinally it is a good idea to check your answers.
\nQuiz to assess matrix addition, subtraction, multiplication and multiplication by scalar, determinants and inverses, solving a system of simultaneous equations.
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