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Line 36: − + − + − a & b \\+ − c & d − \end{bmatrix} \begin{bmatrix} − e \\ f − \end{bmatrix}=\begin{bmatrix} − ae+bf \\ − ce+df \end{bmatrix} − It is important to note that matrix multiplication is not [[commutative]]: in general, <math>AB \neq BA</math> for two matrices <math>A</math> and <math>B</math>. This has important consequences in [[quantum mechanics]].+ − To see why matrix multiplication works the way it does, we will use [[suffix notation]]. Consider first forming the product of two matrices, <math>AB</math>, which is itself a matrix. Then form the product <math>ABx</math>. Matrix multiplication is associative, so we can consider this as either <math>(AB)x</math> or <math>A(Bx)</math>. In suffix notation,+ − − <math> − \sum_j(AB)_{ij}x_j = \sum_k A_{ik}(Bx)_k=\sum_{j,k}A_{ik}B_{kj}x_j − </math> − The vector <math>x</math> is arbitrary, so we can therefore deduce the rule for finding the product of two matrices: − − − (AB)_{ij}=\sum_k A_{ik}B_{kj} −
→Multiplication of matrices: rewrite this section
===Multiplication of matrices===
===Multiplication of matrices===
To multiply two matrices, one uses the rule "go along the rows and down the columns". This is best illustrated by a specific example: a matrix times a vector:
To multiply two matrices, use the rule for finding the product of two matrices:
<math>\begin{bmatrix}
<math>
(AB)_{ij}=\sum_k A_{ik}B_{kj}
</math>
</math>
However, not every pair of matrices can be multiplied. In order for matrices <math>A</math> and <math>B</math> to be compatible for multiplication, the number of ''columns'' in <math>A</math> must equal the number of ''rows'' in <math>B</math>. If <math>A</math> is an <math>m \times n</math> matrix and <math>B</math> is an <math>n \times p</math> matrix, the product matrix <math>AB</math> will have <math>m</math> rows and <math>p</math> columns.
Matrix multiplication is [[associativity|associative]]. However, matrix multiplication is ''not'' [[commutativity|commutative]]. That is, it is possible for <math>AB \neq BA</math>.
<math>
</math>
===Matrix concepts===
===Matrix concepts===