Matrices
Collection of interactive activities related to matrices.
Table of Contents
- The Basics - Definitions, Operations,...
- Systems of Linear Equations
- Solving 2x2 and 3x3 Systems Using Cramer's Rule
- Transformations
- Transformations of Triangles and Matrices
- Matrices and Linear Transformations
- Conics
- Conics and Euclidean Invariants
Solving 2x2 and 3x3 Systems Using Cramer's Rule
A linear system is called:[br][list][*][b][i]consistent[/i][/b] if it has at least one solution. In particular it is [i][b]independent [/b][/i]if it has exactly one solution and [i][b]dependent [/b][/i]if it has infinitely many solutions.[/*][*][i][b]inconsistent[/b][/i] if it has no solutions.[/*][/list]
Cramer's Rule
Given a system [math]Ax=b[/math] of [math]n[/math] linear equations for [math]n[/math] unknowns, if the determinant [math]D[/math] of the matrix [math]A[/math] is nonzero, the system has a unique solution, given by:[br][math]x_1=\frac{D_1}{D}[/math], [math]x_2=\frac{D_2}{D},\ldots x_n=\frac{D_n}{D}[/math] [br]where [math]D_1,D_2,\ldots D_n[/math] are the determinants of the matrices obtained by replacing the [math]i[/math]-th column of [math]A[/math] with the column vector [math]b[/math].
Write the following system in matrix form, then determine whether it can be solved using the Cramer's Rule.[br][math]\begin{cases} 3x + 5y =9 \\ 7x-2y =10 \end{cases}[/math]
Which value of parameter [math]k[/math] makes the following system independent?[br][math]\begin{cases} x - y -3z =8 \\ 3x+ky-z =4 \\2x+3y+4z =-4 \end{cases}[/math][br]
Solve the following system using the Cramer's Rule.[br][math]\begin{cases} -x + 5y=2 \\ 7x-2y=0 \end{cases}[/math]
If applicable, solve the system [math]Ax=b[/math] using Cramer's Rule, given:[br][math]A=\begin{pmatrix} 3 & 2 \\ 2 & 1 \end{pmatrix}[/math] and [math]b=\begin{pmatrix} 6 \\ -2 \end{pmatrix}[/math]
Solve the following system using the Cramer's Rule.[br][math]\begin{cases} x + y+z=6 \\ x-2y-z=-6 \\ 3x+3y-z=6 \end{cases}[/math]
If applicable, solve the system [math]Ax=b[/math] using the Cramer's Rule, given:[br][math]A=\begin{pmatrix} 1 & 1 & -1 \\ 4 & -1 & -5 \\ 1 & -4 & -2 \end{pmatrix}[/math] and [math]b=\begin{pmatrix} 5 \\ 6 \\ -4 \end{pmatrix}[/math]
Transformations of Triangles and Matrices
Drag the vertices of the given triangle and observe how the coordinates matrix changes accordingly.[br][br]Choose a predefined transformation or create your custom one using the appearing sliders.[br][br]The coordinates of the transformed triangle can be obtained by multiplying the transformation matrix by the given triangle's coordinates matrix.
Ready, Set, Practice!
Given the transformation matrix [math]T=\begin{pmatrix}[br]1 & 2\\[br]2 & 1[br]\end{pmatrix} [/math], that maps [math]\left(x,y\right)\rightarrow\left(x',y'\right)[/math] , write the equations of the transformation, then find the images of the points [math]O=\left(0,0\right),A=\left(1,2\right)[/math] and [math]B=\left(-1,1\right)[/math].[br][br]Use the app above to check your results, by selecting the [i]Custom[/i] option and setting the matrix using the displayed sliders.
Select [i]Dilation[/i] from the list of transformations in the app above.[br]Observe the measures of the areas displayed, and how they change when you drag the slider. [br][br]In particular, check the values obtained when [math]k=\pm1[/math] and [math]k=\pm2[/math].[br][br]What is the relationship between the areas of the given triangle and its image?[br]Does this relationship depend on the dilation ratio [math]k[/math]?
Select [i]Dilation[/i] from the list of transformations in the app above, and set the ratio [math]k=-1[/math].[br][br]Describe the relative position of the given triangle and its image.[br]Observe the transformation matrix.[br][br]Now, without modifying the given triangle, select [i]Rotation[/i], and apply a 180° rotation to the given triangle.[br][br]What do you notice? Can you generalize this?[br]
Conics and Euclidean Invariants
Enter the coefficients of the conic in the input boxes, calculate the Euclidean invariants and explore the graph of the corresponding conic.[br][br]You can [i]zoom in/out[/i] if the graph is not fully visible in the Graphics View.