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# Matrix Multiplication Calculator

## This calculator provides a detailed solution that explains how to multiply two matrices. It is possible to multiply two matrices only if the number of columns of the first matrix is equal to the number of rows of the second

An m ร n matrix is a table of numbers with m rows and n columns. Matrix elements are denoted as aij, where i is the row number, j is the column number.

Dimension of matrices
A: ร
B: ร

Matrix 1
A =
Matrix 2
B =

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### Examples of matrix multiplication

Multiply two matrices of dimensions 3 ร 2 and 2 ร 3
C = A ร B =
 4 1 0 5 2 -3
ร
 0.45 12 -9 3 1.4 0
=
 4.8 49.4 -36 15 7 0 -8.1 19.8 -18

Solution

Given two matrices
A =
 a11 a12 a21 a22 a31 a32
where,
a11 = 4
a12 = 1
a21 = 0
a22 = 5
a31 = 2
a32 = -3
B =
 b11 b12 b13 b21 b22 b23
where,
b11 = 0.45
b12 = 12
b13 = -9
b21 = 3
b22 = 1.4
b23 = 0

You can multiply two matrices only if the number of columns of matrix A is equal to the number of rows of matrix B

When multiplying the l ร m matrix A by the m ร n matrix B, we get the l ร n matrix C

C =
 a11 a12 a21 a22 a31 a32
ร
 b11 b12 b13 b21 b22 b23
=
 c11 c12 c13 c21 c22 c23 c31 c32 c33

Matrix element with index Cij is found by the formula

c11 = a11 โ b11 + a12 โ b21 = 4 โ 0.45 + 1 โ 3 = 4.8

c12 = a11 โ b12 + a12 โ b22 = 4 โ 12 + 1 โ 1.4 = 49.4

c13 = a11 โ b13 + a12 โ b23 = 4 โ (-9) + 1 โ 0 = -36

c21 = a21 โ b11 + a22 โ b21 = 0 โ 0.45 + 5 โ 3 = 15

c22 = a21 โ b12 + a22 โ b22 = 0 โ 12 + 5 โ 1.4 = 7

c23 = a21 โ b13 + a22 โ b23 = 0 โ (-9) + 5 โ 0 = 0

c31 = a31 โ b11 + a32 โ b21 = 2 โ 0.45 + (-3) โ 3 = -8.1

c32 = a31 โ b12 + a32 โ b22 = 2 โ 12 + (-3) โ 1.4 = 19.8

c33 = a31 โ b13 + a32 โ b23 = 2 โ (-9) + (-3) โ 0 = -18
C =
 4.8 49.4 -36 15 7 0 -8.1 19.8 -18
Go to calculator
Multiply two 2 ร 2 and 2 ร 4 matrices
C = A ร B =
 7 1 0 3
ร
 -5 6 2 9 12 37 1 0
=
 -23 79 15 63 36 111 3 0

Solution

Given two matrices
A =
 a11 a12 a21 a22
where,
a11 = 7
a12 = 1
a21 = 0
a22 = 3
B =
 b11 b12 b13 b14 b21 b22 b23 b24
where,
b11 = -5
b12 = 6
b13 = 2
b14 = 9
b21 = 12
b22 = 37
b23 = 1
b24 = 0

You can multiply two matrices only if the number of columns of matrix A is equal to the number of rows of matrix B

When multiplying the l ร m matrix A by the m ร n matrix B, we get the l ร n matrix C

C =
 a11 a12 a21 a22
ร
 b11 b12 b13 b14 b21 b22 b23 b24
=
 c11 c12 c13 c14 c21 c22 c23 c24

Matrix element with index Cij is found by the formula

c11 = a11 โ b11 + a12 โ b21 = 7 โ (-5) + 1 โ 12 = -23

c12 = a11 โ b12 + a12 โ b22 = 7 โ 6 + 1 โ 37 = 79

c13 = a11 โ b13 + a12 โ b23 = 7 โ 2 + 1 โ 1 = 15

c14 = a11 โ b14 + a12 โ b24 = 7 โ 9 + 1 โ 0 = 63

c21 = a21 โ b11 + a22 โ b21 = 0 โ (-5) + 3 โ 12 = 36

c22 = a21 โ b12 + a22 โ b22 = 0 โ 6 + 3 โ 37 = 111

c23 = a21 โ b13 + a22 โ b23 = 0 โ 2 + 3 โ 1 = 3

c24 = a21 โ b14 + a22 โ b24 = 0 โ 9 + 3 โ 0 = 0
C =
 -23 79 15 63 36 111 3 0
Go to calculator
Multiply two matrices of dimensions 1 ร 3 and 3 ร 2
C = A ร B =
 2 -7 0
ร
 3 6 4 1 0 8
=
 -22 5

Solution

Given two matrices
A =
 a11 a12 a13
where,
a11 = 2
a12 = -7
a13 = 0
B =
 b11 b12 b21 b22 b31 b32
where,
b11 = 3
b12 = 6
b21 = 4
b22 = 1
b31 = 0
b32 = 8

You can multiply two matrices only if the number of columns of matrix A is equal to the number of rows of matrix B

When multiplying the l ร m matrix A by the m ร n matrix B, we get the l ร n matrix C

C =
 a11 a12 a13
ร
 b11 b12 b21 b22 b31 b32
=
 c11 c12

Matrix element with index Cij is found by the formula

c11 = a11 โ b11 + a12 โ b21 + a13 โ b31 = 2 โ 3 + (-7) โ 4 + 0 โ 0 = -22

c12 = a11 โ b12 + a12 โ b22 + a13 โ b32 = 2 โ 6 + (-7) โ 1 + 0 โ 8 = 5
C =
 -22 5
Go to calculator