矩阵的逆与行列式计算示例

GeoGebra 命令及参数

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