# Matrix Multiplication Calculator

2x2
3x3
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6x6

### Basic Information

Definition: Suppose $$A=[a_{ik}]$$ and $$B=[b_{kj}]$$ are matrices such that the number of columns of A is the same as the number of rows of B; say, A is $$m \times p$$ matrix and B is a $$p \times n$$ matrix. Then the product AB is the $$m \times n$$ matrix whose ij-entry is obtained by multiplying the i-th row of A by the j-th column of B. ie:

$$\begin{bmatrix} a_{11} & \dots & a_{1p}\\ . & \dots & .\\ a_{i1} & \dots & a_{ip}\\ . & \dots & .\\ a_{m1} & \dots & a_{mp} \end{bmatrix}$$ $$\begin{bmatrix} b_{11} & \dots & b_{1j} & \dots & b_{1n}\\ . & \dots & . & \dots & .\\ . & \dots & . & \dots & .\\ b_{p1} & \dots & b_{pj} & \dots & b_{pn} \end{bmatrix} =$$ $$\begin{bmatrix} c_{11} & \dots & c_{1n}\\ . & \dots & .\\ . & c_{ij} & .\\ . & \dots & .\\ c_{m1} & \dots & c_{mn}\\ \end{bmatrix}$$
where
$$c_{ij}=a_{i1}b_{1j} + a_{i2}b_{2j} + \dots + a_{ip}b_{pj}$$.

Few theorems that may facilitate calculations of matrix multiplication:

a) $$(AB)C = A(BC)$$;
b) $$A(B+C) = AB + AC$$;
c) $$k(AB) = (kA)B = A(kB)$$
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