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Using properties of matrix operations

Sal determines which of a few optional matrix expressions is equivalent to the matrix expression A*B*C. This is done using what we know about the properties of matrix addition and multiplication. Created by Sal Khan.

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  • orange juice squid orange style avatar for user Berend Baas
    Don't we get into problems here as A^2 has the dimensions of A * A, and A * A might not be defined (2x3 * 2x3)? Can we do algebraic manipulation even if we don't yet know if the operation is defined for all matrices?
    (8 votes)
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  • blobby green style avatar for user konstantinos1993
    How do we know that A(BC+A) is equal to ABC + A^2 and not BCA + A^2 or BAC + A^2?
    (6 votes)
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    • piceratops ultimate style avatar for user m.v.
      Matrix multiplication is not (in general) commutative for multiplication. That is, if you change the order of multiplication (AB to BA) you don't always get the same answer. (In fact, sometimes, because of the dimensions of the matrices, you cannot even find the reversed product.) So you can't treat these A's, B's, and C's the same as you did in Algebra I because they are representing matrices (not commutative) and not representing real numbers (commutative). You must keep the A, B, and C in the same order.
      (13 votes)
  • piceratops ultimate style avatar for user Kevin George Joe
    at sal talks about a "zero matrix" what does he mean by it?
    (2 votes)
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  • blobby green style avatar for user pjs5858
    Wouldn't the parenthesis force the addition of the ABC clause to the AA clause prior to subtracting the AA clause? I am asking because I do not know if the same Order of Operations applies with matrices.
    (5 votes)
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  • leafers ultimate style avatar for user Tim Wu
    For the very first question (battle school), why does it have to state that ABC are square matricies? Does that affect the answer in any way and if so how?
    (4 votes)
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    • starky tree style avatar for user shruthi
      It matters a lot. For example,if a is a 2*3 matrix A * A might not be defined (2x3 * 2x3) because there is a different no. of columns in the first and rows in the second. But if it is a square matrix, then it will always be definable.
      (3 votes)
  • piceratops tree style avatar for user sirsha dey
    how can we square a matrix ?,like A^2 ?
    (3 votes)
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    • leaf blue style avatar for user jwinder47
      Squaring something (like a matrix or a real number) simply means multiplying it by itself one time: A^2 is simply A x A. So to square a matrix, we simply use the rules of matrix multiplication. (Supposing, of course, that A can be multiplied by itself: not all matrices can be multiplied. So my question to you would be: What kind of matrix can be squared?)
      (2 votes)
  • orange juice squid orange style avatar for user Juan Eduardo Ynsil Alfaro
    How do you know that ABC+A^2 is defined? They could have diferent dimensions
    (2 votes)
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    • leaf green style avatar for user kubleeka
      We're told that A, B, and C are square matrices. We also have that ABC is defined. So A, B, and C must all be the same dimension (if they weren't, one of those multiplications would be undefined).

      When we multiply square matrices of equal dimension, we get another matrix of the same dimension. So ABC must be the same dimension as A, and A^2 must be the same dimension as A.

      So ABC and A^2 must have the same dimension.
      (3 votes)
  • male robot hal style avatar for user Raivat Shah
    Where can we find these types of problems on Khan Academy ?
    (2 votes)
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    • duskpin seedling style avatar for user beestar
      Go to KA, click subjects in the top left hand corner. In the drop-down menu, click Calculus. It should pull up a window with the cord calculus, and right under that will be two options: Explore and Classes. Click Classes, and the click where it says "Precalculus" (the left side). Then you can chose either Explore or Mission. Click mission, and right under the percentage wheel click Show All Skills. There should be a section just for matrices.
      (3 votes)
  • female robot grace style avatar for user lthanedar
    What's a square matrix?
    (2 votes)
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  • male robot hal style avatar for user acha Zeng
    My English is poor. Who can explain me "grade the next wave of student's test"
    Thank you !
    (1 vote)
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Video transcript

Voiceover:In order to get into Battle School cadets have to pass a rigorous entrance exam which includes mathematics. Help Commander Graff grade the next wave of students' tests. The last step of a problem in the matrix multiplication section is the matrix A times B times C where A, B, and C are square matrices. Which of the following candidates have answers that are equivalent to this expression? Select all who are right for any A, B, and C. Select all that apply. I encourage you to pause this video and think about it. Which of these expressions for any square matrices A, B, and C are equivalent to this right over here. I'm assuming you've given a go at it. Let's think through each of them. This one is B, A, C so if they have changed the order and we've already seen that matrix multiplication is not commutative in general and so this will not be true for any square set of matrices A, B, and C, so this is not going to be true. Matrix multiplication is not commutative. Here you have Bernard, who says A times, C times B. We already know that that's going to be the equivalent to A, C, B which once again they've swapped the order between the B and the C, matrix multiplication is not commutative. You can't just swap order and expect to get the same product for any square matrices A, B, and C so we could rule that one out. A times, B, C, so we've already seen matrix multiplication is associative, so this is the same thing as A times B, times C which of course is the same thing as A, B, C. What Caren has right over here, that is right, that is equivalent for any square matrices A, B, and C that is equivalent to A, B, C. Now Ducheval, let's see, now this looks like a bit of a crazy expression but let's think it through a little bit. First of all matrix multiplication, as long as you keep the order right, the distributive property does hold. This first part right over here is equivalent to ... Let me write this down, this one's interesting. We have A times B, C plus A, minus A squared. You can actually distribute this A and I encourage you to prove it for yourself maybe using some two by two matrices for simplicity. This is going to be equal to, this part over here is going to be A, B, C plus A A, A times A which we could write as A squared and then we're going to subtract A squared. These two things are going to cancel out, they're going to end up being the zero matrix and if you take the zero, so these are going to be the zero matrix right over here. If you take the zero matrix and add it to A, B, C you're just going to end up with A, B, C. This one was a little bit tricky, this one actually is equivalent. This one is right and this one is right. Here this is A times B, plus C so this is kind of not-y right over here. They're not even multiplying B and C, so this one's definitely not going to be true for all square matrices A, B, and C.