Number of rows of matrix A:

Number of columns of matrix A:

Number of columns of matrix B:

matrixA,0,0,matrixmatrixA,1,0,matrixmatrixA,2,0,matrixmatrixA,3,0,matrix

matrixA,0,1,matrixmatrixA,1,1,matrixmatrixA,2,1,matrixmatrixA,3,1,matrix

matrixA,0,2,matrixmatrixA,1,2,matrixmatrixA,2,2,matrixmatrixA,3,2,matrix

matrixA,0,3,matrixmatrixA,1,3,matrixmatrixA,2,3,matrixmatrixA,3,3,matrix

matrixB,0,0,matrixmatrixB,1,0,matrixmatrixB,2,0,matrixmatrixB,3,0,matrix

matrixB,0,1,matrixmatrixB,1,1,matrixmatrixB,2,1,matrixmatrixB,3,1,matrix

matrixB,0,2,matrixmatrixB,1,2,matrixmatrixB,2,2,matrixmatrixB,3,2,matrix

matrixB,0,3,matrixmatrixB,1,3,matrixmatrixB,2,3,matrixmatrixB,3,3,matrix

=Solve

How to multiply two 3×3 matrices

The product of two 3×3 matrices A and B is a 3×3 matrix C. The entry in row i and column j of C is the dot product of row i of A with column j of B:

cᵢⱼ = aᵢ₁·b₁ⱼ + aᵢ₂·b₂ⱼ + aᵢ₃·b₃ⱼ

There are nine entries and each needs three multiplications and two additions, so a full 3×3 product is 27 multiplications and 18 additions. The work is simple but easy to get wrong by one slip, which is why it helps to see every entry written out.

Worked example

Let A have rows (1, 0, 2), (−1, 3, 1) and (2, 1, 0). Let B have rows (3, 1, 0), (2, 1, 4) and (1, 0, 2). The columns of B are (3, 2, 1), (1, 1, 0) and (0, 4, 2).

First row of the product: 1·3 + 0·2 + 2·1 = 5, then 1·1 + 0·1 + 2·0 = 1, then 1·0 + 0·4 + 2·2 = 4.

Second row: −1·3 + 3·2 + 1·1 = 4, then −1·1 + 3·1 + 1·0 = 2, then −1·0 + 3·4 + 1·2 = 14.

Third row: 2·3 + 1·2 + 0·1 = 8, then 2·1 + 1·1 + 0·0 = 3, then 2·0 + 1·4 + 0·2 = 4.

The product AB has rows (5, 1, 4), (4, 2, 14) and (8, 3, 4).

Tips for multiplying by hand

Work one row of the result at a time: keep a finger on one row of A and move across the columns of B. Zeros save work, since any term with a zero factor can be skipped.

Remember that AB and BA are usually different, so keep the matrices in the order the problem gives them. Multiplication is still associative, (AB)C = A(BC), and it distributes over addition.

The calculator above is set to two 3×3 matrices and writes out all nine dot products. Entries can be integers, decimals, fractions, complex numbers or variables.

3×3 matrix multiplication worked example

Write the initial matrix 

:

=

1

-1

2

0

3

1

2

1

0

Write the initial matrix 

:

=

The result of multiplying two matrices (

and

) will be a matrix (

) with the same number of rows as in matrix 

 and with the same number of columns as in matrix 

;

Write the initial matrix 

 and mark the elements that we need to find as unknown:

=

×××

To find all the elements of the matrix 

, need to calculate all possible combinations of scalar products of row-vectors of matrix 

 by column-vectors of matrix 

;

To find an element of the matrix 

 with indices 

and

, need to multiply each element of the 

-th row of the matrix 

 by the corresponding element of the 

-th column of the matrix 

 and add the resulting products;

i,j

=

= 1

i,k

·

k,j

// where

i is the row number;

j is the column number;

a is an element of matrix A;

b is an element of matrix B;

c is an element of matrix C;

k is a variable counter, which for each element c_{i,j} will start with a value of 1, increase by 1 at each iteration, and end at a value of m;

m is the number of columns of the matrix A or the number of rows of the matrix B;

=

·

=

1

-1

2

0

3

1

2

1

0

·

=

Answer

c = a · b

SIZE3×3OPS45

Calculators by size and method

Sources

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