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Is matrix multiplication commutative? Associative property of matrix multiplication. Properties of matrix multiplication. Using properties of matrix operations. Next lesson. Current timeTotal duration Google Classroom Facebook Twitter.
Video transcript Voiceover:We know that the multiplication of scalar quantities is commutative. For example, 5 times 7 is the same thing as 7 times 5, and that's obviously just a particular example. I could give many, many more. I could never say it This is the same thing as negative 11 times 3. Or if we wanted to speak in general terms, if I have the scalar a and I multiply it times the scalar b, that's going to be the same thing as multiplying the scalar b times the scalar a.
Now what I want to do in this video is think about whether this property of commutativity, whether the commutative property of multiplication of scalars, whether there is a similar property for the multiplication of matrices, whether it's the case that if I had two matrices, let's say matrix capital A and matrix capital B, whether it's always the case that that product, the resulting matrix here is the same as the product of matrix B and matrix A, just swapping the order.
I encourage you It might be sometimes true, but in order for us to say that matrix multiplication is commutative, that it doesn't matter what order we are multiplying it, we have to figure out is this always going to be true? I encourage you to pause this video and think about that for a little bit. Let's just think through a few things. First of all, let's just think about matrices of different dimensions.
Let's say I have a matrix here. Let's say that matrix A is a, I don't know, let's say it is a 5 by 2 matrix, 5 by 2 matrix, and matrix B is a 2 by 3 matrix. The product AB is going to have what dimensions? If I multiply these two, you're going to get a third matrix.
Let's just call that C for now. You're going to get a third matrix C. What are going to be the dimensions of C? We know, first of all, that this product is defined under our convention of matrix multiplication because the number of columns that A has is the same as the number of rows B has, and the resulting rows and column are going to be the rows of A and the columns of B.
So C is going to be a 5 by 3 matrix, a 5 by 3 matrix. Now what about the other way around? What would B times A be? Once again, I encourage you to pause the video. If you were to take B, let me copy and paste that, and multiply that times A, so I'm really just switching the order of the multiplication so copy and paste. If we take that product right over there, what is that going to be equal to?
What is this? What is this right over here going to be equal to? The first question is, is matrix multiplication even defined for these two matrices? When you look at the number of columns that B has and the number of rows that A has, you see that it actually is not defined, that we have a different number of columns for B and a different number of rows for A. Here, the product is not defined, is not defined, so this immediately is a pretty big clue that this isn't always going to be true.
Here, AB, the product AB is defined, and you'll end up with a 5 by 3 matrix. The product here, BA, isn't even defined. This is already We're already seeing that this is not the case, that order matters when you are multiplying, when you are multiplying matrices. Tuesday's values result from multiplying each Tuesday closing price by its respective number of shares and summing the result for the first portfolio, then doing the same for the second portfolio.
And so on, through the rest of the week. Multiplying a matrix by a scalar is an exception to the dimension and commutative rules. It just operates element-by-element. Matrix division is useful primarily for solving equations, and especially for solving simultaneous linear equations see Solving Simultaneous Linear Equations.
In formal matrix algebra, the solution involves matrix inversion. In general,. In general, matrix A must be a nonsingular square matrix; that is, it must be invertible and it must have the same number of rows and columns. Generally, a matrix is invertible if the matrix times its inverse equals the identity matrix.
To understand the theory and proofs, consult a textbook on linear algebra such as Elementary Linear Algebra by Hill listed in Bibliography. Matrix division is especially useful in solving simultaneous linear equations. Consider this problem: Given two portfolios of mortgage-based instruments, each with certain yields depending on the prime rate, how do you weight the portfolios to achieve certain annual cash flows?
The answer involves solving two linear equations. It is a linear equation because it describes a line in the xy -plane. A system of linear equations is a set of linear equations that you usually want to solve at the same time; that is, simultaneously. A basic principle for exact answers in solving simultaneous linear equations requires that there be as many equations as there are unknowns.
To get exact answers for x and y , there must be two equations. For example, to solve for x and y in the system of linear equations. Matrix algebra represents this system as an equation involving three matrices: A for the left-side constants, X for the variables, and B for the right-side constants.
Solving the system simultaneously means solving for X. In general, you can use matrix algebra to solve any system of linear equations such as. To illustrate, consider this situation. There are two portfolios of mortgage-based instruments, M1 and M2.
An investor holds 10 units of M1 and 20 units of M2. As word equations:. Find the answer by solving two linear equations. The investor should hold Finally, element-by-element arithmetic operations are called operations. Addition and subtraction, and matrix multiplication and division by a scalar, are already array operations so no period is necessary.
When using array operations on two matrices, the dimensions of the matrices must be the same. For example, given vectors of stock dividends and closing prices. Choose a web site to get translated content where available and see local events and offers. Based on your location, we recommend that you select:. Select the China site in Chinese or English for best site performance.
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