Multiplication Matrices :
In the first part we will look in to the multiplication of square matrices. In the next part you will learn to multiply different order matrices (e.g: 2x3 to 3x3).
Here we will multiply a 3x3 matrix (3 rows, 3 columns) to another 3x3 matrix (3 rows, 3 columns).
Matrix A | | Matrix B |
a11 | a12 | a13 |
a21 | a22 | a23 |
a31 | a32 | a33 |
| x |
b11 | b12 | b13 |
b21 | b22 | b23 |
b31 | b32 | b33 |
|
The resulting matrix will be a 3x3 matrix. We will have to calculate each cell of the result matrix separately. Let us assume the result to be X.
Step 1:
To calculate x11
x11 is the cell where first row merges with first column. So in order to calculate the result we will use the first row of Matrix A and first column of Matrix B.
Result X | | Matrix A | | Matrix B |
x11 | x12 | x13 |
x21 | x22 | x23 |
x31 | x32 | x33 |
| = |
a11 | a12 | a13 |
a21 | a22 | a23 |
a31 | a32 | a33 |
| x |
b11 | b12 | b13 |
b21 | b22 | b23 |
b31 | b32 | b33 |
|
Now x11 can be calculated as x11 = a11xb11 + a12xb21 + a13xb31
Step 2:
To calculate x12
x12 is the cell where first row merges with second column. So in order to calculate the result we will use the first row of Matrix A and second column of Matrix B.
Result X | | Matrix A | | Matrix B |
x11 | x12 | x13 |
x21 | x22 | x23 |
x31 | x32 | x33 |
| = |
a11 | a12 | a13 |
a21 | a22 | a23 |
a31 | a32 | a33 |
| x |
b11 | b12 | b13 |
b21 | b22 | b23 |
b31 | b32 | b33 |
|
Now x12 can be calculated as x12 = a11xb12 + a12xb22 + a13xb32
Following the same procedure we will have to calculate values for all cells.
Result Matrix |
a11xb11 + a12xb21 + a13xb31 | a11xb12 + a12xb22 + a13xb32 | a11xb13 + a12xb23 + a13xb33 |
a21xb11 + a22xb21 + a23xb31 | a21xb12 + a22xb22 + a23xb32 | a21xb13 + a22xb23 + a23xb33 |
a31xb11 + a32xb21 + a33xb31 | a31xb12 + a32xb22 + a33xb32 | a31xb13 + a32xb23 + a33xb33 |
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