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Matrix Inverse and Determinant Example

Matrix Inverse and Determinant Example
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2014-10-27 23:28
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In collections UCSS Secondary Math I Unit 6 Lesson 3
This example demonstrates how to compute the additive inverse, determinant, and multiplicative inverse of a matrix. The key mathematical concepts involved include matrix operations, inverse matrices, and determinants. Construction involves using GeoGebra commands such as LaTeX, Corner, Determinant, and TableText to create the matrix and display results. Suitable for high school or introductory college mathematics courses.

GeoGebra Commands & Parameters

  1. LaTeX(steps) → counter
  2. Corner(3) → B
  3. Determinant(A) → determA
  4. LaTeX("\text{Find the additive inverse, }\textcolor{#000099}{–A}.") → text1
  5. LaTeX("\text{The additive inverse of a matrix can be found by multiplying each entry in the}\cr\text{original matrix by }\textcolor{#000099}{–1}\text{; that is, by performing scalar multiplication of a matrix.}") → text2
  6. LaTeX((((("\text{The additive inverse of }\textcolor{#000099}{A=" * TableText[A, "r[]"]) * "}\text{ is }\textcolor{#000099}{-A=") * TableText[{{aneg, bneg}, {cneg, dneg}}, "r[]"]) * "}.")) → text7
  7. LaTeX("\text{Find the determinant of }\textcolor{#000099}{A}.") → text8
  8. LaTeX((((("\text{The determinant of }\textcolor{#000099}{A=" * TableText[A, "r[]"]) * "}\text{ is }\textcolor{#000099}{") * determA) * "}.")) → text14
  9. LaTeX("\text{Find the multiplicative inverse, }\textcolor{#000099}{A^{–1}}\text{, of the original matrix, }\textcolor{#000099}{A}.") → text15
  10. TableText({{(((("\textcolor{#006633}{\frac{" * d) * "}{") * determA) * "}}"), (((("\textcolor{#FF9933}{\frac{-(" * b) * ")}{") * determA) * "}}")}, {(((("\textcolor{#FF9933}{\frac{-(" * c) * ")}{") * determA) * "}}"), (((("\textcolor{#006633}{\frac{" * a) * "}{") * determA) * "}}")}}, "r[]") → Ainv
  11. LaTeX(((" \textcolor{#000099}{" * TableText[{{FractionText[ainv], FractionText[binv]}, {FractionText[cinv], FractionText[dinv]}}, "r[]"]) * "} ")) → Ainv2
  12. LaTeX((((("\textcolor{#000099}{A^{-1}=\frac{1}{" * determA) * "}") * TableText[{{(("\textcolor{#006633}{" * d) * "}"), (("\textcolor{#FF9933}{-(" * b) * ")}")}, {(("\textcolor{#FF9933}{-(" * c) * ")}"), (("\textcolor{#006633}{" * a) * "}")}}, "r[]"]) * "}")) → Ainv0
  13. LaTeX((("\text{Find the additive inverse, the determinant, and the multiplicative inverse of the matrix }\textcolor{#000099}{A=" * TableText[A, "r[]"]) * "}.")) → Problem
  14. TableText({{LaTeX[(("\textcolor{#000099}{A=" * TableText[A, "r[]"]) * "}")], LaTeX["\text{Given matrix}"]}}) → text3
  15. TableText({{LaTeX[(("\textcolor{#000099}{A=" * TableText[A, "r[]"]) * "}")], LaTeX["\text{Given matrix}"]}, {LaTeX[(("\textcolor{#000099}{-A=(-1)" * TableText[A, "r[]"]) * "}")], LaTeX["\text{Multiply the given matrix by }\textcolor{#000099}{–1}\text{ to find the}"]}, {LaTeX[""], LaTeX["\text{additive inverse.}"]}}) → text4
  16. LaTeX("\text{Move the slider to solve the problem.}") → text0
  17. TableText({{LaTeX[(("\textcolor{#000099}{A=" * TableText[A, "r[]"]) * "}")], LaTeX["\text{Given matrix}"]}, {LaTeX[(("\textcolor{#000099}{-A=(-1)" * TableText[A, "r[]"]) * "}")], LaTeX["\text{Multiply the given matrix by }\textcolor{#000099}{–1}\text{ to find the}"]}, {LaTeX[""], LaTeX["\text{additive inverse.}"]}, {LaTeX[(("\textcolor{#000099}{-A=" * TableText[{{(("(-1)(" * a) * ")"), (("(-1)(" * b) * ")")}, {(("(-1)(" * c) * ")"), (("(-1)(" * d) * ")")}}, "r[]"]) * "}")], LaTeX["\text{Multiply each entry by the scalar value, }\textcolor{#000099}{-1}."]}}) → text5
  18. TableText({{LaTeX[(("\textcolor{#000099}{A=" * TableText[A, "r[]"]) * "}")], LaTeX["\text{Given matrix}"]}, {LaTeX[(("\textcolor{#000099}{-A=(-1)" * TableText[A, "r[]"]) * "}")], LaTeX["\text{Multiply the given matrix by }\textcolor{#000099}{–1}\text{ to find the}"]}, {LaTeX[""], LaTeX["\text{additive inverse.}"]}, {LaTeX[(("\textcolor{#000099}{-A=" * TableText[{{(("(-1)(" * a) * ")"), (("(-1)(" * b) * ")")}, {(("(-1)(" * c) * ")"), (("(-1)(" * d) * ")")}}, "r[]"]) * "}")], LaTeX["\text{Multiply each entry by the scalar value, }\textcolor{#000099}{-1}."]}, {LaTeX[(("\textcolor{#000099}{-A=" * TableText[{{aneg, bneg}, {cneg, dneg}}, "r[]"]) * "}")], LaTeX["\text{Simplify.}"]}}) → text6
  19. LaTeX((("\text{The determinant of a }\textcolor{#000099}{2\times 2}\text{ matrix }\textcolor{#000099}{" * TableText[{{"\textcolor{#006633}{a}", "\textcolor{#FF9933}{b}"}, {"\textcolor{#FF9933}{c}", "\textcolor{#006633}{d}"}}, "r[]"]) * "}\text{ can be found using the formula }\textcolor{#000099}{det(A)=}\textcolor{#006633}{ad}\textcolor{#000099}{-}\textcolor{#FF9933}{bc}.")) → text9
  20. TableText({{LaTeX[(("\textcolor{#000099}{det(A)= " * TableText[{{(("\textcolor{#006633}{" * a) * "}"), (("\textcolor{#FF9933}{" * b) * "}")}, {(("\textcolor{#FF9933}{" * c) * "}"), (("\textcolor{#006633}{" * d) * "}")}}, "r||"]) * "}")], LaTeX["\text{Substitute values to find the determinant.}"]}}) → text10
  21. TableText({{LaTeX[(("\textcolor{#000099}{det(A)= " * TableText[{{(("\textcolor{#006633}{" * a) * "}"), (("\textcolor{#FF9933}{" * b) * "}")}, {(("\textcolor{#FF9933}{" * c) * "}"), (("\textcolor{#006633}{" * d) * "}")}}, "r||"]) * "}")], LaTeX["\text{Substitute values to find the determinant.}"]}, {LaTeX[(((((((("\textcolor{#000099}{det(A)=}\textcolor{#006633}{(" * a) * ")(") * d) * ")}\textcolor{#000099}{-}\textcolor{#FF9933}{(") * b) * ")(") * c) * ")}")], LaTeX["\text{Apply the formula for a }\textcolor{#000099}{2\times 2}\text{ determinant.}"]}, {LaTeX[(((("\textcolor{#000099}{det(A)=}\textcolor{#006633}{(" * ad) * ")}\textcolor{#000099}{-}\textcolor{#FF9933}{(") * (b * c)) * ")}")], LaTeX["\text{Multiply.}"]}}) → text12
  22. TableText({{LaTeX[(("\textcolor{#000099}{det(A)= " * TableText[{{(("\textcolor{#006633}{" * a) * "}"), (("\textcolor{#FF9933}{" * b) * "}")}, {(("\textcolor{#FF9933}{" * c) * "}"), (("\textcolor{#006633}{" * d) * "}")}}, "r||"]) * "}")], LaTeX["\text{Substitute values to find the determinant.}"]}, {LaTeX[(((((((("\textcolor{#000099}{det(A)=}\textcolor{#006633}{(" * a) * ")(") * d) * ")}\textcolor{#000099}{-}\textcolor{#FF9933}{(") * b) * ")(") * c) * ")}")], LaTeX["\text{Apply the formula for a }\textcolor{#000099}{2\times 2}\text{ determinant.}"]}, {LaTeX[(((("\textcolor{#000099}{det(A)=}\textcolor{#006633}{(" * ad) * ")}\textcolor{#000099}{-}\textcolor{#FF9933}{(") * (b * c)) * ")}")], LaTeX["\text{Multiply.}"]}, {LaTeX[(("\textcolor{#000099}{det(A)=" * determA) * "}")], LaTeX["\text{Subtract.}"]}}) → text13
  23. TableText({{LaTeX[(("\textcolor{#000099}{det(A)= " * TableText[{{(("\textcolor{#006633}{" * a) * "}"), (("\textcolor{#FF9933}{" * b) * "}")}, {(("\textcolor{#FF9933}{" * c) * "}"), (("\textcolor{#006633}{" * d) * "}")}}, "r||"]) * "}")], LaTeX["\text{Substitute values to find the determinant.}"]}, {LaTeX[(((((((("\textcolor{#000099}{det(A)=}\textcolor{#006633}{(" * a) * ")(") * d) * ")}\textcolor{#000099}{-}\textcolor{#FF9933}{(") * b) * ")(") * c) * ")}")], LaTeX["\text{Apply the formula for a }\textcolor{#000099}{2\times 2}\text{ determinant.}"]}}) → text11
  24. LaTeX((((((("\text{The determinant of a }\textcolor{#000099}{2\times 2}\text{ matrix }\textcolor{#000099}{" * TableText[{{"\textcolor{#006633}{a}", "\textcolor{#FF9933}{b}"}, {"\textcolor{#FF9933}{c}", "\textcolor{#006633}{d}"}}, "r[]"]) * "}\text{, if it exists, can be found using the}\cr\text{formula }\textcolor{#000099}{A^{-1}=\frac{1}{\textcolor{#006633}{ad}\textcolor{#000099}{-}\textcolor{#FF9933}{bc}}") * TableText[{{"\textcolor{#006633}{d}", "\textcolor{#FF9933}{-b}"}, {"\textcolor{#FF9933}{-c}", "\textcolor{#006633}{a}"}}, "r[]"]) * "=\frac{1}{det(A)}") * TableText[{{"\textcolor{#006633}{d}", "\textcolor{#FF9933}{-b}"}, {"\textcolor{#FF9933}{-c}", "\textcolor{#006633}{a}"}}, "r[]"]) * "}.")) → text16
  25. LaTeX("\text{Notice that in order for a matrix to have a multiplicative inverse, its determinant must be nonzero.}") → text17
  26. If(determA ≠ 0, LaTeX[(((((((("\text{Since the determinant of }\textcolor{#000099}{A}\text{ is }\textcolor{#000099}{" * determA) * "}\text{, the multiplicative inverse }\textcolor{#000099}{A^{-1}}\text{ exists and is given by}\cr") * Ainv0) * "\textcolor{#000099}{=}") * Ainv) * "\text{.}\cr\text{This simplifies to }\textcolor{#000099}{A^{-1}=}") * Ainv2) * "\text{, the multiplicative inverse of }\textcolor{#000099}{A}.")], LaTeX[(("\text{Since the determinant of }\textcolor{#000099}{A}\text{ is }\textcolor{#000099}{" * determA) * "}\text{, the multiplicative inverse }\textcolor{#000099}{A^{-1}}\text{ does not exist.}")]) → text18
  27. LaTeX("\text{Use the buttons below to return to the original problem }\cr\text{or randomly generate a similar practice problem.}") → RandomizationText

AI Drawing Prompt Reference

Draw a 2x2 matrix A and its inverse matrix A^-1 and the determinant calculation. First input matrix A, then compute its additive inverse by multiplying each element by -1. Next, calculate the determinant det(A). If det(A) is not zero, compute A^-1. Finally, display matrix A, its additive inverse, the determinant value, and the inverse matrix A^-1.
The prompt above was compiled from the original drawing commands and is for reference only.