Can anyone Explain in a good way... how to multiply matrices together?
542644
Here are a few websites which explain it fairly well...\n\nhttp://www.mai.liu.se/~halun/matrix/\n\nhttp://www.easycalculation.com/matrix/matrix-multiplication.php\n\nThat one has a matrix calculator online. Good luck!
584514
Use 'Gaussian elimination' to obtain a basis for the span of matrix columns or rows.\n\nhttp://en.wikipedia.org/wiki/Gaussian_elimination
869931
You need to show that the only way to make a combination of them sum to zero is for all the multipliers to be zero.  For instance, say you have three vectors x, y, and z .  Then you need to show that if \n\nAx + By + Cz = 0 \n\nis true, then A, B, and C must ALL be equal to zero.  That is, the only way for the above equation to be true is if all the coefficients are zero.  If you can find a set of A, B, and C that satisfy the above equation, then the vectors are NOT linearly independent.
1126894
This normally means that you are going to be using Gaussian (or Gauss-Jordan) reduction, where ther eare three "moves" that are allowed.\n\n1) You can switch any two rows in the matrix.\n2) You can multiply any row in the matrix by a non-zero scalar.\n3) You can add a multiple of one row to another.\n\nUsing these three "moves," the goal is to reduce the matrix into a usable form (the echelon form, for example).
273264
Lets call this matrix A.\n[ 2 0 4 ]\n[ 3 1 5 ]\n[ -1 1 -2 ] = A\n\nNow we need to solve the equation AV = I\nthen V is the inverse matrix.  Lets write this out\n\n[ 2 0 4 ] ` ` ` ` [ 1 0 0 ]\n[ 3 1 5 ] ` ` ` ` [ 0 1 0 ]\n[ -1 1 -2 ] V = [ 0 0 1 ]\n\nNow just multiply elementary matrices (row operations) to the left hand side of both sides of the equation until the left-hand entity is the identity.  Multiply the following matrix on the left-hand-side both sides [[1/2 0 0][0 1 0][0 0 -1]], that is replace the first row with 1/2 times the first row, and replace the third row with -1 times the third row:\n\n[ 1 0 2 ] ` ` ` ` [ 1/2 0 0 ]\n[ 3 1 5 ] ` ` ` ` [ 0 ` 1 0 ]\n[ 1 -1 2 ] V = [ 0 ` 0 -1 ]\n\napply [[1 0 0][[-3 1 0][0 0 1]] both sides, that is replace second row with second row plus negative-three times first row\n\n[ 1 0 2 ] ` ` ` ` [ 1/2 0 0 ]\n[ 0 1 -1 ] ` ` ` ` [ -3/2 1 0 ]\n[ 1 -1 2 ] V = [ 0 ` 0 -1 ]\n\nmultiply both sides on the left by the matrix [[1 0 0][0 1 0][ -1 0 1]], that replace third row with third row plus negative one times first row\n\n[ 1 0 2 ] ` ` ` ` [ 1/2 0 0 ]\n[ 0 1 -1 ] ` ` ` ` [ -3/2 ` 1 0 ]\n[ 0 -1 0 ] V = [ -1/2 ` 0 -1 ]\n\nmultiply by [[1 0 0][0 0 -1][0 1 0]], that is place opposite of row 3 for row 2 and row 2 for row 3\n\n[ 1 0 2 ] ` ` ` ` [ 1/2 0 0 ]\n[ 0 1 0 ] ` ` ` ` [ 1/2 0 1 ]\n[ 0 1 -1 ] V = [ -3/2 1 0 ]\n\nnow apply [[1 0 0][0 1 0][ 0 -1 1]] both sides, that is replace third row with third row plus -1 times second row\n\n[ 1 0 2 ] ` ` ` ` [ 1/2 0 0 ]\n[ 0 1 0 ] ` ` ` ` [ 1/2 0 1 ]\n[ 0 0 -1 ] V = [ -2 ` 1 -1 ]\n\napply [[ 1 0 2][0 1 0][0 0 -1]] both sides, replace row 1 with row 1 plus 2 times row 3, and replace row 3 with negative 1 times row 3\n\n[ 1 0 0 ] ` ` ` ` [ -3/2 2 -2 ]\n[ 0 1 0 ] ` ` ` ` [ 1/2 0 1 ]\n[ 0 0 1 ] V = [ 2 -1 1 ]\n\nNow, since the left hand item is the identity, the inverse matrix is\n\n[ -3/2 2 -2 ]\n[ 1/2 0 1 ]\n[ 2 ` -1 1 ]\n\nThe product of all of the elementary operation matrices specified is also this matrix, and the reader can verity that AV = VA = I
1069671
yep...scalars multiply vectors component by component just like you have shown.  you can also show it by plotting out the original vectors, scaling them appropriately, adding them, and finding the resukltant components.
458991
No, but if you go to List mode you can enter your matrix as a list and manipulate it.  Check out the owner's manual for help on how to do it.  I've included a link to it.  Hope this helps.
595375
divide the first row by 2\n1 1/2\n1  -1\n\nSubtract the first row from the second row:\n1  1/2\n0  -3/2\n\nMutiply the second row by -2/3\n1  1/2\n0   1\n\nSubtract 1/2 of the second row from the first row:\n1  0\n0  1\n\nAnd you have the identity matrix
738212
you need to set up a matrix equation.  \nthe rows should look like this:\n\n1      ,   -2    ,     1    ,      / -8\n3      ,   -3    ,    -2    ,     /-5\n-2     ,   -3     ,    3      ,    /5\n\nOnce you set this up follow the rules of matrices to get this to a unit matrix.  Once you have done this the three numbers on the right will become your answers for x, y and z.  x being which ever row has the 1 in the first column.  y being which ever row has the 1 in the second column.  z being which ever row has the 1 in the third column.
201316
This website may help you.  Good luck.\n\nhttp://www.sosmath.com/matrix/matrix.html
194771
1. Write down a couple of Jordan blocks.\n\n2. Write down their squares, cubes, maybe even a fourth power.\n\n3. Gaze at each set of identity matrix, J, J^2, J^3, ... until you see a simple linear combination of the matrices that sums to the all zero matrix.  That is your minimal polynomial.\n\n4. Generalize.\n\n5. Now it is obvious. :-)\n\nAlternative:\n\n1.  Google for some combination of:\n\nminimal polynomial Jordan block\n"minimal polynomial" "Jordan block"\nJordan block minimal polynomial\n"Jordan block" "minimal polynomial"\n\n2. Read answer.\n\n3. Too bad, it still isn't obvious. :-(
221530
Add them together and divide by 5.\n\nWow. I could seriously have you people do my homework for me. Every night. I'll type out the problems and see how you all do. :P JP
398921
To make an augmented matrix, just take your coefficients and put them in a matrix, then draw a vertical line and place the constants on the right side, like this:\n\n| -2 0 9 | 2 |\n| 0 -2 9 | 22 |\n| 3 5 -2 | 10 |\n\nFill in 0's for any variables that aren't there (i.e. there's no y in the first equation, so there's a 0 in its place in the matrix).
924127
just multiply the side of the cube 3 times
339689
Yes, it's quite unclear, but I will try to guess.\n\n4a+b=38\n2a+3b=24\n\nDid you mean that we can multiply all but one equations in system in such way that coefficients with one variable ("a" for example) will be the same?\n\nExample: Multiplying second equation by 2.\n4a+b=38\n4a+6b=48\n\nNow we can subtract one equation from the other(s) and chosen variable ("a" in our case) will present only in one equation.\n\n4a+b=38\n5b=10\n\nNow we can get b from the second...\n\nb=2\n\n... and substitute it into the first.\n\n4a + 2 = 38\n4a = 36\na = 9\n\n\nThis method is the simplest example of Gaussian elimination (see attached article) and is often use to solve systems of linear equations.
954567
Check out Markov Chain's.  You will find many uses for matrices.
74442
I recommend purlplemath.com.  i'm not sure how helpful it is overal, but i like how they explain matrix multiplication <http://www.purplemath.com/modules/mtrxmult.htm>.  you can also do a web search for the subject you have a problem with, there are plenty of sites.  and your teacher should love to help you.