Matrix Multiplication Tutorial

Matrix Multiplication Tutorial

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
a11a12a13
a21a22a23
a31a32a33
x
b11b12b13
b21b22b23
b31b32b33

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
x11x12x13
x21x22x23
x31x32x33
=
a11a12a13
a21a22a23
a31a32a33
x
b11b12b13
b21b22b23
b31b32b33

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
x11x12x13
x21x22x23
x31x32x33
=
a11a12a13
a21a22a23
a31a32a33
x
b11b12b13
b21b22b23
b31b32b33

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 + a13xb31a11xb12 + a12xb22 + a13xb32a11xb13 + a12xb23 + a13xb33
a21xb11 + a22xb21 + a23xb31a21xb12 + a22xb22 + a23xb32a21xb13 + a22xb23 + a23xb33
a31xb11 + a32xb21 + a33xb31a31xb12 + a32xb22 + a33xb32a31xb13 + a32xb23 + a33xb33

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