Matrix multiplication Definition and 108 Threads

In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the second matrix. The resulting matrix, known as the matrix product, has the number of rows of the first and the number of columns of the second matrix. The product of matrices A and B is denoted as AB.Matrix multiplication was first described by the French mathematician Jacques Philippe Marie Binet in 1812, to represent the composition of linear maps that are represented by matrices. Matrix multiplication is thus a basic tool of linear algebra, and as such has numerous applications in many areas of mathematics, as well as in applied mathematics, statistics, physics, economics, and engineering.
Computing matrix products is a central operation in all computational applications of linear algebra.

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  1. R

    Matrix Multiplication and Rank of Matrix

    Dear Forum, I have one question on matrix multiplication. Suppose there are 2 matrices - A = 1 -1 0 0 2 -1 2 0 -1 B = 1 1 2 and AB = 0 (Zero Matrix) if B not a zero-matrix, then rank(A) is less than s, where s is the dimension of B. I wanted to...
  2. E

    Matrix multiplication preserve order Block matrix

    Hello, If I have block matrices A,B,C and D all of which are non singular would this relationship hold; my main concern is preserving order of matrix multiplication: if ADB=C then AD=B^-1C D=B^-1CA^-1 D^-1 = (BC^-1A) Also is it okay to assume the inverse of a block...
  3. Y

    How Do You Construct a Matrix to Select Specific Elements from Another Matrix?

    I have a small problem where I remember I learned it in the past. But now I can't seem to recall how to do it. I searched on google, but I think even though it is probably really easy, I may not have worded it right to get the references I want. So I decided to ask here. Here is the problem...
  4. James889

    Why is my C matrix multiplication program not working properly?

    Hello, I am trying to write a program that multiplies together two 3x3 matrices. However my program does not really work as intended, and i get some really strange results. Any ideas? #include <stdio.h> #include <stdlib.h> int main() { int result_matrix[3][3]; int matrixA[3][3] = {...
  5. E

    The basis of n x n matrices with matrix multiplication

    Hi All, I recently came across the interesting notion of constructing the minimal set of nxn matrices that can be used as a basis to generate all nxn matrices given that matrix multiplication, and addition and multiplication by scalar are allowed. Is there a way to construct an explicit set...
  6. H

    Matrix Multiplication and Algebraic Properties of Matrix Operations

    1) If A = [aij] is an n x n matrix, the trace of A, Tr(A), is defined as the sum of all elements on the main diagonal of A, Tr(A) = the sum of (aii) from i=1 to n. Show each of the following: a) Tr(cA) = cTr(A), where c is a real number b) Tr(A+B) = Tr(A) + Tr(B) c) Tr(A(Transpose)) = Tr(A)...
  7. M

    NxN matrix multiplication; commutativity

    Homework Statement Explicitly show, using numbers, that in general NxN matrix multiplication is not commutative. Show this for all N > 1. If I make a matrix A, 2x2, 1 2 3 4 and multiply by matrix B, also 2x2 5 6 7 8 I get AB =/=BA If I multiply a new matrix A, 3x3, 1...
  8. R

    What is the meaning of \textbf{nn} in matrix multiplication?

    I can't figure out what an author means by this expression: \textbf{n} \cdot \textbf{e} \cdot (\textbf{I} - \textbf{nn}) and \left(\textbf{u} - \textbf{U}\right) \cdot (\textbf{I} - \textbf{nn}) Here, all I know is that \textbf{u} and \textbf{U} are vectors of length 3. \textbf{n} is a...
  9. G

    Mathematica [Mathematica] Ordered matrix multiplication

    Hello everyone, I have a list, let's call it L1, of length N. Each element of this list is a square matrix. I would like to : 1/ Apply MatrixExp[] to each element of L1 (I know how to do that) 2/ Multiply each element of the subsequent list _in an ordred fashion_ i.e. from element N to...
  10. A

    Concept of Matrix Multiplication

    Here is my question(It is not a homework problem.) What does it mean to multiply two matrices? I know how to do the operation but what is the concept behind it? I understand dot products and vectors. Matrix multiplication can be seen as a dot product operation of row vector of vector A to...
  11. K

    Proof on matrix multiplication

    Homework Statement A is an mxn matrix and C is a 1xm matrix. Prove that CA=Sum of (C sub j)*(A sub j) from j=1 to m. Where A sub j is the jth row of A Sorry for the messy problem statement, I couldn't figure out the summation notation on here. Homework Equations The Attempt at a...
  12. R

    Multiplicative Identity under Matrix Multiplication

    I've been asked by my professor to identify a group of singular matrices. At first, I did not think this was possible, since a singular matrix is non-invertible by definition, yet to prove a groups existence, every such singular matrix must have an inverse. It has been brought to my...
  13. I

    What is the correct way to multiply matrices?

    Hey, When trying to multiply the 2 attached matrixes (row X column) I get a much more terms then the attached answer. C is for cos and s for sin. What am I doing wrong? Regards, i.l
  14. J

    Matrix Multiplication Explained: Why & What's the Pro?

    if A is [A1|A2|A3|...|An] and C is 1 column matrix given by [C1|C2|C3|...|Cn]T then AC=C1A1+C2A2+...+CnAn.. If D is a diagonal matrix then AD = [D11A1+D22A2+...+DNnAn]. please explain why it happens tht way and whts the prof?
  15. J

    Linear Algebra - Matrix Multiplication

    | 1 1 1 | = matrix A | 1 2 3 | | 1 4 5 | How do I find a 3x3 matrix B, excluding the identity or zero matrix, such that AB = BA?
  16. P

    Recurrence relation in matrix multiplication

    Homework Statement T(n)=b n<=2 8T(n/2)+e(n^(2)) n>2 Homework Equations The Attempt at a Solution O((n^log)2^(7))
  17. E

    Matrix multiplication and the dot product

    Homework Statement Suppose that A is an n x n matrix such that A(Transpose)A=I. Let x be any vector in R^n. Show that llAxll=llxll; that is, multiplication of x by A produces a vector Ax having the same length as x. Homework Equations Sqrt(x(transpose)x)=llxll The Attempt at a...
  18. J

    What is the Proof for Matrix Multiplication with Invertible Matrices?

    Homework Statement I'm doing a proof in which I need to show: given that AX = 0, AVX=0 where V is invertible. Also, given that AVY = 0, then AY = 0. Homework Equations The Attempt at a Solution I can't remember from the previous course I took how to do this. I know that I can...
  19. L

    Solving Matrix Multiplication: A*B^T with A=2,1;5,3 and B=1,-1,1;-1,1,1

    Homework Statement Find A*B^T, where A= 2 1 5 3 And B= 1 -1 1 -1 1 1 Homework Equations n/a The Attempt at a Solution The problem is, is that if I get the transpose of B, I will end up with a 3x2 matrix. Isn't this impossible to multiply by a 2x2...
  20. N

    I'm stuck on a matrix multiplication problem

    Homework Statement If AB = BA and p is a nonnegative integer, show that (AB)^p = (A^p)(B^p) Homework Equations N/A The Attempt at a Solution I would love to attempt a solution, but I don't even know where to begin. I guess my first question is, how can I generalize matrices so...
  21. J

    When is matrix multiplication commutative

    Can somebody please explain to me when matrices commute? I've read that it's when they are diagonal with the same dimensions (and also scalar multiples?), but I don't understand why this is.
  22. J

    Inductive proof for matrix multiplication problem

    Homework Statement Compute [1 1]^n [ 1] This is what my book has. I am assuming the blank means 0 (please tell me if this is not the case). Homework Equations None really (matrix multiplication) The Attempt at a Solution I did n = 1, 2, 3, 4 and found a patten. It looks...
  23. J

    Mathematica Question (Matrix Multiplication)

    I am trying to compute the following, \prod_{j=0}^{N-1}\left[\hat{I}+\hat{M(j)}\left(\frac{T}{N}\right)\right] where \hat{I}, \hat{M(j)} are matrices. My problem is that Mathematica interprets this product as element-wise with respect to the matrices, but I of course want it to use matrix...
  24. Saladsamurai

    Show that Matrix Multiplication is Associative

    Homework Statement Show that (AB)C=A(BC) I am just trying to do this to try to gain some experience with problems like this. I saw in my text that they did a similar example for distributivity using the definition of matrix multiplication, so I thought I could use that approach. The Attempt...
  25. 1

    MATLAB Matlab Matrix Multiplication Query in an fft application

    Hey everyone, first of all id just like to clarify this isn't a homework or coursework related problem, its part of my personal MatLab learning for future years. I obtained a nice little siganls and systems question off a friend, it looked fairly simple but I've come across a problem which i...
  26. C

    MATLAB Matlab Code for Matrix Multiplication

    Hihi, Let's say I have the linear equations ,A 2x +3y +4z 5x+6y-7z 5x-4y+3z then I can always write as a matrix multiplication A= B*x matrix B = 2 3 4 5 6 -7 5 -4 3 times matrix x x y z Then., how can I tell MATLAB to express in the matrix multiplication form so...
  27. J

    Composition of Linear Transformation and Matrix Multiplication

    Theorem 2.15: Let A be an m x n matrix with entries from F. Then the left-multiplication transformation L_A: F^n --> F^m. Furthermore, if B is any other m x n matrix ( with entries from F ) and B and D are the standard ordered bases for F^n and F^m, respectively, then we have the following...
  28. G

    Why is matrix multiplication necessary for representing linear transformations?

    What would be the proof for matrix multiplication?...or just an explanation as to why its done the way its done.
  29. J

    Composition of Linear Transformation and Matrix Multiplication

    I am reading (theorem 2.14) from a textbook, and don't understand how g = Tf and (#1) line of reasoning. The theorem and proof is as follows: Theorem 2.14: Let V and W be finite-dimensional vector spaces having ordered bases B and C, respectively, and let T: V-->W be linear. Then, for...
  30. U

    Vector subspace as the space of solutions to matrix multiplication

    Given a subset W of a vector space V = F^n (for some field F), prove that W is the subspace of solutions of the matrix equation AX = 0 for some A.
  31. N

    Matrix Multiplication Properties for 2x2 Matrices

    Where A = a 2*2 matrix, is the following true: (A^n)(A^m) = (A^m)(A^n) Thanks in advance
  32. R

    Matrix Multiplication and Inverses

    Homework Statement Show that A and Inverse(I+A) commute (where I is the identity matrix). Homework Equations Inverse(Inverse(A))=A Inverse(AB)=Inverse(B)*Inverse(A) The Attempt at a Solution My solution assumes the existence of the inverse of A. A*Inverse(I+A) =...
  33. M

    Matrix multiplication: Communicative property.

    Matrix multiplication: Commutative property. Hello, First time poster. I have got a question about commutative property of matrix multiplication. Literature says that matrix multiplication is communicative only when the two matrices are diagonal. But, I have a situation with an...
  34. T

    How to Multiply 4x4 Matrices Using a 3x3 Calculator?

    i am given two matrices 4X4 which i need to multiply by one another but in my calculator i can multiply matrices only 3X3 how can i use this maximum 3X3 matrices multiplication in order to calculate the multiplication of two 4X4 matrices ?? (i know how to multiply matrices in theory...
  35. T

    Matrix multiplication question

    i am given two matrices 4X4 which i need to multiply by one another but in my calculator i can multiply matrices only 3X3 how can i use this maximum 3X3 matrices multiplication in order to calculate the multiplication of two 4X4 matrices ?? (i know how to multiply matrices in...
  36. Shaun Culver

    Proving the Formula for Matrix Multiplication | Homework Statement & Equations

    Homework Statement Prove the formula. Homework Equations Matrix multiplication: (\text{AB})_{i \,j}=\sum _{k=1}^n a_{i \,k}b_{k \,j} The Attempt at a Solution I do not know how to "prove" the formula for arbitrary values of k and n.
  37. A

    Prove: A(BC)=(AB)C for matrix multiplication

    Homework Statement Prove the following theorem: A(BC)=(AB)C. Homework Equations http://en.wikipedia.org/wiki/Matrix_multiplication#Ordinary_matrix_product" The Attempt at a Solution Let A be of order m by n, B be of order n by p, and C be of order p by q. Then...
  38. C

    Matrix multiplication of an nxn matrix is the scaling

    Geometrically, matrix multiplication of an nxn matrix is the scaling, and rotation of a vector in n dimensions true? So when you find the inverse of a matrix, what you're actually doing is finding a transformation such that in the 'transformed space' the vector is a unit vector. If the inverse...
  39. R

    Solving A^7 Using Matrix Multiplication: A 3x3 Example

    Homework Statement I can't figure out how to latexa 3X3 matrix so here's my ghetto method A = 0 0 -1 0 2 0 0 0 1 What is Homework Equations I'm trying to find A^{7} The Attempt at a Solution I'm assuming there's some type of shortcut to get...
  40. J

    Why Is Matrix Multiplication Defined the Way It Is?

    Something that has bothered me in my linear algebra class was that I learned a lot of techniques but didn't learn why they worked, or what they were useful for. One of the things is this: why is matrix multiplication so useful in the way it's defined, and not in any other way? Of all the ways...
  41. S

    Successive matrix multiplication

    On a given day of a flu epidemic, a given percentage Y of the population is ill and (1-Y) is healthy. The probability of remaining healthy on the next day is a, and that of remaining sick is B. The question is, what percentage will be ill after a given number of days as a function of B,a and Y...
  42. M

    Simplifying Matrix Multiplication: A Beginner's Guide

    Matrix multiplication confuses me. How, for example, would I multiply these matrices: a b c d e f g h i x r s t u v w x y z ? Thanks!
  43. T

    Can These Matrices Be Multiplied?

    What is \left(\begin{array}{ccc}1&1&2\\0&2&1\\1&0&3\end{array}\right)\left(\begin{array}{cc}1&1\\3&3\end{array}\right)? I'm so confuse because the first matrix is 3 columns matrix and the second matrix is 2 rows matrix. Thank you
  44. C

    How Can XY = 1 and YZ = 1 Prove X = Z?

    Homework Statement Suppose one has n×n square matrices X, Y and Z such that XY = 1and Y Z = 1. Show that it follows that X = Z. The Attempt at a Solution Now I know if the equatoins had been XY and ZY I would do this: XY=ZY -> XY-ZY=0 -> Y(X-Z)=0 -> X-Z=0 -> X=Z I was wondering if...
  45. V

    Matrix Multiplication of \delta_{ij}v_j = v_i

    Homework Statement Show by matrix multiplication, \delta_{ij}v_j = v_i The Attempt at a Solution I'm having trouble understanding how to do this, because I'm under the impression that v_j is a row vector, which can't be multiplied by a 3x3 matrix which \delta_{ij} is; or am I horribly...
  46. P

    A short one on matrix multiplication

    Let A and B be real invertible n x n matrices so that B = (AT)-1. Show that Bm = (I - B1A1T)(AmT)+, where B1 = [b1, ..., bm], Bm = [bm+1, ..., bn], A1 = [a1, ..., am], Am = [am+1, ..., an]. Any pointers on how one would go about proving the above? I'm fresh out of ideas. I'm not...
  47. A

    Infinite-dimensional matrix multiplication

    We know that infinite-dimensional matrix multiplication in general isn't asociative. But, is there any criteria when asociativity is valid? thanks in advance.
  48. P

    Interpretation of matrix multiplication?

    Matrix multiplication is clearly defined but is there a tangible or physical interpretation for it? Or is it just abstract formalism? I am thinking about each column of the matrix as vectors so matrix multiplication with two 2by2 matrices is about multiplying 4 vectors in a certain way. The...
  49. mattmns

    Lin Alg - Matrix multiplication (Proof by contrapositive)

    Hello, here is the question my book is asking: Let A, B be two m x n matricies. Assume that AX = BX for all n-tuples X. Show that A = B. ------- So I decided to try and prove the contrapositive, which is (unless I am mistaken): If A \neq B, then there is some X such that AX \neq BX Proof...
  50. quasar987

    Why was matrix multiplication defined the way it is?

    What was the (historical) motivation for defining the rules of matrix multiplication the way it is?
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