Special Sub-Topic: Enter the Matrix
|How do you add matrices?|
Add the corresponding components of the matrices. Adding matrices is one of the simplest operations you can perform. Subtraction works the same way: subtract the corresponding components of each matrix.
Note: Matrices must have the same dimension in order for you to add/subtract them.
|How do you multiply matrices?|
Take the dot product of the rows of the first with the columns of the second. If you think of the first entry of the product matrix as entry 1,1 (row, column), the dot product of the 1st row of the first matrix with the 1st column of the second matrix will be entry 1,1. The dot product of the 1st row of the first matrix with the 2st column of the second matrix will be entry 1,2. The dot product of the 2st row of the first matrix with the 1st column of the second matrix will be entry 2,1 and so on until you have exhausted all row-column combinations.
Note: The matrices don't have to be the same dimension to multiply them. In order to multiply them, the number of columns in the first matrix has to be the same as the number of rows in the second matris. The dimension of the resulting product matrix will always have the same number of rows as the first matrix and the same number of columns as the second matrix.
|How do you divide matrices?|
You can't divide matrices. You can't divide matrices. The way you would get around this is by multiplying by the inverse of the matrix. If you think about simple numbers, 12/4 is the same as 12*(1/4). Finding the inverse of a matrix will be discussed later in this quiz.
|Row-reduction, also know as Gauss-Jordan Elimination, can be used for which of the following?|
All of these (Solving systems of equations using an augmented matrix, Finding the determinant of the matrix, Finding the rank of the matrix). You can multiply rows by scalars and add them together to get new rows to replace the old ones with the goal of achieving the identity matrix. An augmented matrix is a matrix with more columns than rows. The most common augmented matrix is one where there is one more column than row, and you place the coefficients of the equations in the "body" of the matrix and the numbers the equations equal in the augmented part of the matrix (the last column). Once you have obtained the identity matrix in the "body", whatever is in the last column will be your solutions to the system.
Finding the determinant of a matrix involves using row reduction to obtain an upper triangular matrix (all terms below the diagonal are zero). at which point you can multiply the diagonal terms to obtain the determinant. You then have to multiply the determinant by a factor determined by the way you performed row operations. While row-reducing, each row-scaling multiplies the determinant by the scalar, and each row swap multiplies the determinant by negative one. This is not a very practical way to find the determinant but is useful every now and then.
Finding the rank involves row reducing until you have a leading one in every row. The rank is the number of leading ones you have.
|Finding the inverse of a matrix involves augmenting the matrix by the identity matrix and performing row reduction until|
the "body" part of the matrix looks like the identity matrix. Once the "body" part of the matrix becomes the identity matrix, the augmented part of the matrix (which originally was the identity matrix) will be a random-looking matrix. This matrix is the inverse of the one you started with.
|This matrix operation tells you whether or not the matrix is invertible. It is also useful for telling you area change during linear transformations, finding eigenvectors, and using Cramer's Rule to solve equations.|
determinant. If you have a 2x2 matrix
|c d|, the determinant is ad-bc. For a matrix
|a b c|
|d e f|
|g h i|, the determinant is (aei+bfg+cdh)-(ceg+afh+bdi).
As an aside, the discriminant is the expression under the radical in the quadratic formula, the derivative indicates the slope of the tangent line of a function, and demonstrative is an English adjective meaning "Serving to manifest or prove" (American Heritage Dictionary).
|So now that you know the matrix operations, how about a few examples before we move on? Find the determinant of:
[3 -5] [-2 4]
[8 -1]+[-7 0]|
0. Adding the components of the matrices yields:
|1 -1|. The determinant is (1*-1)-(-1*1)=(-1)-(-1)=0
As a side note, a determinant of zero means that the matrix is not invertible.
|Solve this system:
x=1, y=-2, z=0. The matrix is:
[2 -4 6|10]
[0 1 1|-2]
[1 2 3|-3]. The 10, -2, -3 behind the lines is the augmented part. The row reduction yields:
[2 -4 6|10] => [1 -2 3| 5] => [1 0 3| 1] => [1 0 0| 1]
[0 1 1|-2] => [0 1 0|-2] => [0 1 0|-2] => [0 1 0|-2]
[0 4 0|-8] => [0 1 1|-2] => [0 0 1| 0] => [0 0 1| 0].
Thus x=1, y=-2, z=0.
Sorry for any misalignment on this answer. It's hard writing out matrices in this format.
|For any given matrix A, any vector x such that Ax=0 is said to be in the ________ of A, and all linearly independent column vectors of A are said to make up the _________ of A.|
kernel; image. Linearly independent columns means that no combination of one or more previous columns could compose a later column. For instance, the columns of the matrix:
[1 2 3]
[4 5 6]
[7 8 9]
are not linearly independent because the third column can be made up by 2*(v2)-1*(v1).
|For any given 3x3 matrix the dimension of the kernel (number of vectors in the kernel) is the same as the dimension of the image (number of vectors in the image).|
never true. The rank-nullity theorem states that the dimension of the kernel (aka nullity) plus the dimension of the image (aka rank) must equal the number of columns in that matrix. The reason for this is since any non-linearly independent vector can be expressed as a sum of the other vectors, a non-trivial relationship can be set up thus enabling you to find a nonzero vector in the kernel of the matrix. Getting back to the problem, since there are 3 columns in a 3x3 matrix, the rank and the nullity could never be equal because no two equal whole numbers add up to three.
|For a matrix A there exists one or more eigenvectors, which, when A is applied to them, are only scaled by a constant, but not rotated. What does "eigen" mean in German?|
proper. It means proper. Eigenvectors are useful for solving things such as discrete or continuous dynamical systems, viewing quadratic forms, etc.
|Each eigenvector has a corresponding eigenvalue. What does the eigenvalue tell you?|
The number by which the eigenvector is scaled. When a matrix A is applied to an eigenvector v, v is scaled by a constant factor, known as the eigenvalue.
The equation Av=λv is often used, where A is a matrix, v is an eigenvector, and λ is a constant.
|Which of the following is an eigenvalue-eigenvector pair of this matrix:
[ 1 2]
[ 3 0].
The answers will be in the form a,[b][c], where a is the eigenvalue and
is the eigenvector.|
-2, [-3]. This is an eigenvector because:
[1 2]*[ 2]=[-4]
[3 0] [-3] [ 6], which is -2 times the initial vector.
The zero vector works, but the definition of eigenvector states that it has to be a non-zero vector.
 is an eigenvector of this matrix, but its eigenvalue is 3, not -3.
|5x2+4xy+5y2=84 could be described as what? (Hint: This problem has to do with the use of eigenvectors.)|
an ellipse rotated 45 degrees from the x-axis. You can easily eliminate the hyperbola choice because all of the terms are positive. You would need at least one negative term for it to be a hyperbola. Breaking this down into a matrix A with the x² and y² coefficients on the diagonal, and splitting the middle term into two parts, we get:
[2 5]. We can obtain eigenvectors for this matrix by finding two vectors in the kernel of (λI-A) where λ is an eigenvalue. We obtain eigenvectors:
[-1] . These act as our new principle axes, x and y, respectively. It is easy then to see that this is an ellipse rotated 45-degrees becaue the lines built through the points (1,1) and (1,-1) (with the origin) make a 45-degree angle with the x and y axes. Taking the problem further, the graph 5x²+4xy+5y²=84 is equivalent to 3x²+7y²=84, rotated 45 degrees clockwise. Note that 3 and 7 are the eigenvalues of the matrix A, above. Cool, huh?
|There is an isolated town of population 1000. There are two little family-run businesses in that town. Warrick's General Store (WGS) has 40% of the town business. Johnson's Town Shoppe (JTS) has 60% of the town business. JTS can offer slightly lower prices though, so every month 10% of the WGS shoppers start shopping at JTS; however, 5% of JTS customers return to WGS because of friendlier service. All is fine until a Walmart moves in. Because Walmart pays its laborers almost nothing, they can sell the same goods for cheaper, so naturally people start to go to Walmart. Every month, 20% of the customers from WGS go to Walmart, and 25% of the customers from JTS go to Walmart. However, because people begin to feel guilty for shopping at a gross example of globalization, every month 5% of the people at Walmart go back to WGS and 10% of the people at Walmart go back to JTS. Assuming that: (a) The population of shoppers stays constant at 1000; (b) the competition between JTS and WGS remains even after Walmart moves in and; (c) People switch shopping locations simultaneously (as opposed to sequentially), approximately how many people will remain at each store after a large number of months have passed?|
Walmart: 607; WGS: 143; JTS: 250. This is another eigenvector problem. Given that every month:
1. Walmart gains 20% from WGS and 25% from JTS
2. WGS gains 5% from Walmart and 5% from JTS
3. JTS gains 10% from Walmart and 10% from WGS
You can set up a 3x3 matrix showing this.
[.85 .20 .25]
[.05 .70 .05]
[.10 .10 .70] where column 1 is Walmart, column 2 is WGS and column 3 is JTS.
Because this is a standard transitional matrix (the columns sum to 1) one of the eigenvalues must be 1. Using the ker(λI-A) method, find that the corresponding eigenvector is:
These numbers sum to 28, so Walmart will take 17/28 of the business, or 607 shoppers, WGS will take 4/28 of the business, or 143 shoppers, and JTS will take 7/28 of the business, or 250 shoppers. Looking at the original numbers and percentages, you can see that this makes sense. (You can also see how big stores like Walmart can ruin family-run businesses.)
Well, that's all for matrices on this quiz. I hope you either enjoyed it, did well, or learned from it. (Hopefully more than one of those applied to you.)
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