Step-by-step explanation:
Approximately how many times greater is 8.5×108 than 3.4×105 ? A 2.5 B 250 C 2,500 D 25,000
2 answers:
Solution:
we have been asked to find that , how many times one number is to the other given numbers.
To do this we will divide the larger number by the smaller number and the resulting number will be the answer of the question.
Here the larger number is
The smaller number is
Now lets divide as follows
Hence the correct option is C.
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Let's work with 2-by-2 matrices so we're on the same page. The ideas will work for any appropriate matrices.
From the rule of matrix multiplication, we see:
![\left[\begin{array}{cc}a_{11} & a_{12} \\a_{21} & a_{22} \end{array}\right] \left[\begin{array}{cc}b_{11} & b_{12} \\b_{21} & b_{22} \end{array}\right] = \left[\begin{array}{cc} a_{11}b_{11} + a_{12}b_{21} & a_{11}b_{12} + a_{12}b_{22} \\ a_{21}b_{11} + a_{22}b_{21} & a_{21}b_{12} + a_{22} b_{22} \end{array}\right]](https://tex.z-dn.net/?f=%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7Da_%7B11%7D%20%26%20a_%7B12%7D%20%5C%5Ca_%7B21%7D%20%26%20a_%7B22%7D%20%5Cend%7Barray%7D%5Cright%5D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7Db_%7B11%7D%20%26%20b_%7B12%7D%20%5C%5Cb_%7B21%7D%20%26%20b_%7B22%7D%20%5Cend%7Barray%7D%5Cright%5D%20%3D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D%20a_%7B11%7Db_%7B11%7D%20%2B%20a_%7B12%7Db_%7B21%7D%20%26%20a_%7B11%7Db_%7B12%7D%20%2B%20a_%7B12%7Db_%7B22%7D%20%5C%5C%20a_%7B21%7Db_%7B11%7D%20%2B%20a_%7B22%7Db_%7B21%7D%20%26%20a_%7B21%7Db_%7B12%7D%20%2B%20a_%7B22%7D%20b_%7B22%7D%20%5Cend%7Barray%7D%5Cright%5D%20)
As you noted, we see the columns of B contributing to the rows of C. The question is, why would we ever have defined matrix multiplication this way?
Here's a nontraditional way of feeling this connection. We can define matrix multiplication as "adding multiplication tables." A multiplication table is made by starting with a column and a row. For example,
We then fill this table in by multiplying the row and column entries:
![\begin{array}{ccc} {} & [1] & [2] \\ 1| &1 & 2 \\ 2| & 2 &4 \end{array}](https://tex.z-dn.net/?f=%5Cbegin%7Barray%7D%7Bccc%7D%20%7B%7D%20%26%20%5B1%5D%20%26%20%5B2%5D%20%5C%5C%201%7C%20%261%20%26%202%20%5C%5C%202%7C%20%26%202%20%264%20%5Cend%7Barray%7D)
It's then reasonable to say that given two matrices A and B, we can construct multiplication tables by taking the columns of A and pairing them with the rows of B:
![\left[\begin{array}{cc}a_{11} & a_{12} \\a_{21} & a_{22} \end{array}\right] \left[\begin{array}{cc}b_{11} & b_{12} \\b_{21} & b_{22} \end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7Da_%7B11%7D%20%26%20a_%7B12%7D%20%5C%5Ca_%7B21%7D%20%26%20a_%7B22%7D%20%5Cend%7Barray%7D%5Cright%5D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7Db_%7B11%7D%20%26%20b_%7B12%7D%20%5C%5Cb_%7B21%7D%20%26%20b_%7B22%7D%20%5Cend%7Barray%7D%5Cright%5D%20)
![= \begin{array}{cc} {} & \left[\begin{array}{cc} b_{11} & b_{12}\end{array} \right]\\ \left[\begin{array}{c} a_{11} \\ a_{21} \end{array} \right] \end{array} +\begin{array}{cc} {} & \left[\begin{array}{cc} b_{21} & b_{22}\end{array} \right]\\ \left[\begin{array}{c} a_{12} \\ a_{22} \end{array} \right] \end{array}](https://tex.z-dn.net/?f=%3D%20%5Cbegin%7Barray%7D%7Bcc%7D%20%7B%7D%20%26%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D%20b_%7B11%7D%20%26%20b_%7B12%7D%5Cend%7Barray%7D%20%5Cright%5D%5C%5C%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7D%20a_%7B11%7D%20%5C%5C%20a_%7B21%7D%20%5Cend%7Barray%7D%20%5Cright%5D%20%5Cend%7Barray%7D%20%2B%5Cbegin%7Barray%7D%7Bcc%7D%20%7B%7D%20%26%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D%20b_%7B21%7D%20%26%20b_%7B22%7D%5Cend%7Barray%7D%20%5Cright%5D%5C%5C%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7D%20a_%7B12%7D%20%5C%5C%20a_%7B22%7D%20%5Cend%7Barray%7D%20%5Cright%5D%20%5Cend%7Barray%7D)
![= \left[\begin{array}{cc} a_{11} b_{11} & a_{11} b_{12} \\ a_{21} b_{11} & a_{21} b_{12} \end{array} \right] + \left[\begin{array}{cc} a_{12} b_{21} & a_{12} b_{22} \\ a_{22} b_{21} & a_{22} b_{22} \end{array} \right]](https://tex.z-dn.net/?f=%3D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D%20a_%7B11%7D%20b_%7B11%7D%20%26%20a_%7B11%7D%20b_%7B12%7D%20%5C%5C%20a_%7B21%7D%20b_%7B11%7D%20%26%20a_%7B21%7D%20b_%7B12%7D%20%5Cend%7Barray%7D%20%5Cright%5D%20%2B%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D%20a_%7B12%7D%20b_%7B21%7D%20%26%20a_%7B12%7D%20b_%7B22%7D%20%5C%5C%20a_%7B22%7D%20b_%7B21%7D%20%26%20a_%7B22%7D%20b_%7B22%7D%20%5Cend%7Barray%7D%20%5Cright%5D)
Adding these matrices together, we get the exact same expression as the traditional definition.
From the rule of matrix multiplication, we see:
As you noted, we see the columns of B contributing to the rows of C. The question is, why would we ever have defined matrix multiplication this way?
Here's a nontraditional way of feeling this connection. We can define matrix multiplication as "adding multiplication tables." A multiplication table is made by starting with a column and a row. For example,
We then fill this table in by multiplying the row and column entries:
It's then reasonable to say that given two matrices A and B, we can construct multiplication tables by taking the columns of A and pairing them with the rows of B:
Adding these matrices together, we get the exact same expression as the traditional definition.