Matrix Multiplication Calculator

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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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