dimension of a matrix
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A matrix is a rectangular arrangement or array of numbers often called elements. Here are two examples of matrices. Notice how both matrices have six elements, but the arrangement is quite different. And this is why its so important to understand the dimensions of a matrix. The size or dimension, m by n, of a matrix identifies how many rows and columns a specific matrix has. The numbers of rows is m, and the number of columns is n, where rows go left and right, and columns go up and down. So looking at this first matrix, it has one, two rows and one, two, three columns. And therefore, the dimensions of this first matrix would be two by three. Comparing this to the second matrix, notice how this matrix has one, two, three rows and one, two columns. Therefore, the dimensions would be three by two. So this is why its so important that we understand the dimensions of a matrix. Notice if we consider a specific element in the matrix, for example, this element here. Notice how its identified by a sub two comma three because the position of this element is row two, column three.
If we compare this to this element in the second matrix, we have a sub three comma two because this element is in row three, column two. Lets take a look at several more examples. We can see from these four examples its common to identify a matrix using a capital letter. So if we look at matrix A, notice how it has one, two, three, four rows and one column. Therefore, the dimensions would be four by one. And often when identifying a matrix using a capital letter, as we see here, well often write capital A with the subscript for the dimensions, which in this case would be four by one. And because this matrix has a single column, it can also be called a column matrix. If we look at matrix B, notice how it has one row and one, two, three columns. Therefore, the dimensions would be one by three. Because it has a single row, its often called a row matrix. And if we identify this using the variable B, well often write the dimensions as a subscript, so we would have one by three, like this. Looking at matrix C, notice how
we have one, two, three rows and one, two, three columns. So the dimensions would be three by three. Because the number of rows and columns are the same, this is called a square matrix, where, again, if we use the capital letter C, we would have C with a subscript of three by three. And for our last example, for matrix D, notice how we have one, two rows and one, two, three, four columns. So the dimensions would be two by four. And again, if we use the capital letter D to identify this matrix, we would put the dimensions as a subscript, as we see here. Now, I do want to end with a couple of reasons why knowing the dimensions of a matrix is so important. As we already briefly discussed, the dimensions of a matrix must be known to identify a specific element in a matrix. For example, if we were to identify the element in row two, column one, we have to understand that that means were looking for the element in row two, column one. So heres row two and heres column one, so a sub two comma one would be
to nine. Another reason is to add or subtract matrices, the dimensions must be the same. Notice how this first matrix is a two by two matrix, and the second matrix is also a two by two matrix, so because these dimensions are the same, we would be able to add these matrices. If the dimensions arent the same, we could not add or subtract the matrices. And then finally, to multiply matrices, the number of columns in the first matrix must be the same as the number of rows in the second matrix in order to perform matrix multiplication. So for example, this first matrix has two rows and one column, so its a two by one matrix. The second matrix has one row and three columns, so its a one by three matrix. And so we can only multiply these matrices if this number, the number of columns in the first matrix, and this number, the number of rows in the second matrix, are the same. And since they are, we can perform this multiplication. If they werent, we couldnt. We will talk about how to perform matrix operations in future videos. Hope you found this helpful.
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