Note that the definition of [I][I] stipulates that the multiplication must commute, that is, it must yield the same answer no matter in which order multiplication is done. What matrix has this property? It is important to confirm those multiplications, and also confirm that they work in reverse order as the definition requires. There is no identity for a non-square matrix because of the requirement of matrices being commutative.
The reason for this is because, for two matrices to be multiplied together, the first matrix must have the same number of columns as the second has rows. Privacy Policy. Skip to main content. Search for:. Introduction to Matrices. Learning Objectives Describe the parts of a matrix and what they represent.
Key Takeaways Key Points A matrix whose plural is matrices is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. Matrices can be used to compactly write and work with multiple linear equations, that is, a system of linear equations. Key Terms element : An individual item in a matrix row vector : A matrix with a single row column vector : A matrix with a single column square matrix : A matrix which has the same number of rows and columns matrix : A rectangular array of numbers, symbols, or expressions, arranged in rows and columns.
Learning Objectives Practice adding and subtracting matrices, as well as multiplying matrices by scalar numbers. Key Takeaways Key Points When performing addition, add each element in the first matrix to the corresponding element in the second matrix. When performing subtraction, subtract each element in the second matrix from the corresponding element in the first matrix.
Addition and subtraction require that the matrices be the same dimensions. The resultant matrix is also of the same dimension. Scalar multiplication of a real Euclidean vector by a positive real number multiplies the magnitude of the vector without changing its direction.
Key Terms scalar : A quantity that has magnitude but not direction. Learning Objectives Practice multiplying matrices and identify matrices that can be multiplied together.
The product of a square matrix multiplied by a column matrix arises naturally in linear algebra for solving linear equations and representing linear transformations.
Key Terms matrix : A rectangular arrangement of numbers or terms having various uses such as transforming coordinates in geometry, solving systems of linear equations in linear algebra and representing graphs in graph theory. Donate Login Sign up Search for courses, skills, and videos. Math Precalculus Matrices Multiplying matrices by matrices. Intro to matrix multiplication.
Multiplying matrices. Practice: Multiply matrices. Next lesson. Current timeTotal duration Google Classroom Facebook Twitter. Video transcript We're given two matrices over here, matrix E and matrix D. And they ask us, what is ED, which is another way of saying what is the product of matrix E and matrix D? Just so I remember what I'm doing, let me copy and paste this.
And then I'm going to get out my little scratch pad. So let me paste that over here. So we have all the information we needed. And so let's try to work this out. So matrix E times matrix D, which is equal to-- matrix E is all of this business. So it is 0, 3, 5, 5, 5, 2 times matrix D, which is all of this. So we're going to multiply it times 3, 3, 4, 4, negative 2, negative 2. Now the first thing that we have to check is whether this is even a valid operation. Now the matrix multiplication is a human-defined operation that just happens-- in fact all operations are-- that happen to have neat properties.
Now the way that us humans have defined matrix multiplication, it only works when we're multiplying our two matrices. So this right over here has two rows and three columns. So it's a 2 by 3 matrix.
And this has three rows and two columns, it's 3 by 2. The first way is to multiply a matrix with a scalar. This is known as scalar multiplication. The second way is to multiply a matrix with another matrix. That is known as matrix multiplication. To multiply a scalar with a matrix, we simply take the scalar and multiply it to each entry in the matrix. Let's do an example.
Question 1 : Calculate 2 A 2A 2 A if. The question is asking us to find out what 2 A 2A 2 A is. In other words, we are finding. Question 2 : Calculate 0 A 0A 0 A if. Again, we are trying to find 0 A 0A 0 A. This means that we will be looking for the answer to. The matrix will be oddly shaped, but the concept remains the same. We will still multiple the scalar 0 to each entry in the matrix. Doing so gives us:. Notice that all the entries in the matrix are 0.
This is known as a zero matrix that is 3 x 2. Now that we are very familiar with scalar multiplication , why don't we move on to matrix multiplication? Dot product also known as vector multiplication is a way to calculate the product of two vectors. For example, let the two vectors be:. How would I multiply these two vectors?
Simply just multiply the corresponding entries, and add the products together. In other words,. So we get a single value from multiplying vectors. However, notice how that the two vectors have the same number of entries.
What if one of the vectors has a different number of entries than the other? For example, let. There is a problem here. The first three entries have corresponding entries to multiply with, but the last entry doesn't. So what do we do here? The answer is we cannot do anything here. This just means we cannot calculate the dot product of these two vectors.
So in conclusion, we cannot find the dot product of two vectors that have different numbers of entries. They must have the same number of entries. So what was the point of learning the dot product? Well, we will be using the dot product when we multiply two matrices together.
When multiplying a matrix with another matrix, we want to treat rows and columns as a vector. More specifically, we want to treat each row in the first matrix as vectors, and each column in the second matrix as vectors.
Now we are going to treat each row and column we see here as a vector. Notice here that multiplying a 2 x 2 matrix with another 2 x 2 matrix gives a 2 x 2 matrix. In other words, the matrix we get should have 4 entries. How do we exactly get the first entry? Well, notice that the first entry is located on the first row and first column.
Thus, the first entry will be. How do we get the second entry this time? Well, notice that the location of the second entry is in the first row and second column. Thus, the second entry will be. Now we are going to use the same strategy to look for the last two entries. This gives us:. Now we are done! This is what we get when we are multiplying 2 x 2 matrices. In general, the matrix multiplication formula for 2 x 2 matrices is.
Now the process of a 3 x 3 matrix multiplication is very similar to the process of a 2 x 2 matrix multiplication.
Again, why don't we do a matrix multiplication example? If we are to keep locating all the entries and doing the dot product corresponding to the rows and columns, then we get the final result. We are done! Notice that the bigger the matrices are, the more tedious matrix multiplication becomes. This is because we have to deal with more and more numbers! In general, the matrix multiplication formula for 3 x 3 matrices is.
So far we have multiplied matrices with the same dimensions. In addition, we know that multiplying two matrices with the same dimension gives a matrix of the same dimensions. But what happens if we multiply a matrix with different dimensions?
How would we know the dimensions of the computed matrix? First, we need to see multiplying the matrices gives you a defined matrix.
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