Matrices
Collection of interactive activities related to matrices.
Table of Contents
- The Basics - Definitions, Operations,...
- Matrices - Basic Terminology
- Square Matrix Calculator
- 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
Matrices - Basic Terminology
Use the app to learn the basic terminology related to matrices, then complete the exercises below the app to check your understanding.
Ready, Set, Practice!
A [i]magic square[/i] is a square array of natural numbers, all different from each other, such as the sums of the elements of each row, each column and the two diagonals are all the same value, that is the "[i]magic constant[/i]" of the square. The [i]order[/i] of the square is the number of elements along one side.[br][br]Create a magic square of order 3.[br]What is your magic constant?
A matrix has the following elements:[br][math]a_{11}=2[/math], [math]a_{22}=-1[/math], [math]a_{12}=3[/math], [math]a_{21}=0[/math].[br][br]Write the matrix.[br]Is this a square or rectangular matrix? What is the order (size) of the matrix?
Write the expression for a square matrix of order 4 such that[br][math]a_{ij}=a_{ji}[/math] [math]\forall i\in\left[1,n\right][/math] , [math]\forall j\in\left[1,n\right][/math][br][br](Which value of [math]n[/math] should you consider for this problem?)
Write the expression for a square matrix of order 4 such that[br][math]a_{ij}=i+j[/math] [math]\forall i\in\left[1,n\right][/math] , [math]\forall j\in\left[1,n\right][/math][br][br](Which value of [math]n[/math] should you consider for this problem?)
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.