To add two Matrices, two Vectors, or a Matrix and a scalar, use the syntax $A+B$.  In the case where one of $A$ or $B$ is a Matrix and the other is a scalar, the scalar is interpreted as a diagonal Matrix of appropriate dimensions, with that scalar value along the diagonal.

To multiply two Matrices, a Matrix and a Vector, a Matrix or Vector and a scalar, or to compute the dot product of two Vectors, use the syntax $A·B$.  This "." operator is non-commutative, so it does not rearrange the orders of non-scalar terms.

If $A$ and $B$ are numbers (including complex and extended numerics such as infinity and undefined), then $A·B=AB$, that is, the '.' operator extends the '*' operator in this case.

If $A$ and $B$ are Vectors with the same orientation (that is, both are row Vectors or both are column Vectors) and dimension, then $A·B$ is computed as their dot product.

If one of $A$ and $B$ is a Matrix or a Vector, and the other is a Matrix, Vector, or constant and the previous case does not apply, then their product is computed as the relevant algebraic operation, without reordering.  That is, the '.' operator implements non-commutative multiplication.

The '.' operator is n-ary, meaning that expressions such as $A·B·C$ are interpreted as expected.

Note:  In Maple,  '.' can be interpreted as a decimal point (for example, $3.7$), as part of a range operator (for example, $x..y$), or as the (non-commutative) multiplication operator.  To distinguish between these three circumstances, Maple uses the following rule.

Any dot that is not part of a range operator (more than one '.' in a row) and not part of a number is interpreted as the non-commutative multiplication operator.

Note that the interpretation of the phrase "not part of a number" depends on whether you are using 1-D or 2-D input mode.  In 1-D input mode, interpretation proceeds from left to right, and a dot following a number will be interpreted as a decimal point unless that number already contains a decimal point.  In 2-D input mode, interpretation is carried out on the expression as a whole, and because spaces and juxtaposition can be interpreted as multiplication, a dot which is immediately preceded or followed by a number is always interpreted as a decimal point.

For example, in 1-D input mode, 3.4 is a number, 3. 4 is an error and 3 .4 and 3 . 4 return 12.  3. .4 is 12. and 3..4 is a range.

In 2-D input mode, 3.4 is a number, 3. 4 and 3 .4 are errors and 3 . 4 returns 12.  3. .4 is an error and 3..4 is again a range.  (All of the errors shown by these examples are due to the rule that a number cannot appear as the right-hand operand of an implicit multiplication operation. In such cases, use of explicit multiplication (*) can avoid the error.  See also 2-D Math Details for more information.)

It is an error to use the "*" operator between operands that are both Matrices or Vectors. To multiply a scalar, x, and a Matrix or Vector, you can use $x$ * $A$ or $A$ * $x$.  There is a small, but important, difference between $x·A$ and $xA$ in this case, where $x$ is a scalar and $A$ is a Matrix or Vector.  Namely, if $x$ is a symbolic expression (not a constant), then $xA$ performs the componentwise multiplication, while $x·A$ returns unevaluated.

If $A$ is a square Matrix, then its (integer) powers can be computed using the syntax $A^n$, where $n$ is an integer.  If $n$ is negative, $A^n={\left(A^{\left(-1\right)}\right)}^{\left(-n\right)}$, where $A^{\left(-1\right)}$ is the inverse of $A$ (if it exists).

The transpose of a Matrix or Vector is obtained by the special syntax $A^{+}$ or the Transpose(A) command.

Similarly, the Hermitian transpose of a Matrix or Vector is obtained by the special syntax $A^{*}$ or the HermitianTranspose(A) command.

The cross product of two 3-D Vectors is computed using the syntax $v\&xw$.

$\mathrm{with}\left(\mathrm{Student}:-\mathrm{LinearAlgebra}\right)\:$

$A≔⟨⟨1\,2⟩\left|⟨3\,4⟩\right|⟨5\,6⟩⟩$

$A≔\left[\begin{array}{ccc}1& 3& 5\\ 2& 4& 6\end{array}\right]$

$A+1$

$\left[\begin{array}{ccc}2& 3& 5\\ 2& 5& 6\end{array}\right]$

$B≔⟨⟨a\,b⟩|⟨c\,d⟩⟩$

$B≔\left[\begin{array}{cc}a& c\\ b& d\end{array}\right]$

$B·A$

$\left[\begin{array}{ccc}a+2c& 3a+4c& 5a+6c\\ b+2d& 3b+4d& 5b+6d\end{array}\right]$

$A·B$

Error, (in LinearAlgebra:-Multiply) first matrix column dimension (3) <> second matrix row dimension (2)

$A^{\mathrm{\%T}}$

$\left[\begin{array}{cc}1& 2\\ 3& 4\\ 5& 6\end{array}\right]$

$B^{-1}$

$\left[\begin{array}{cc}\frac{d}{ad-bc}& -\frac{c}{ad-bc}\\ -\frac{b}{ad-bc}& \frac{a}{ad-bc}\end{array}\right]$

$B^{\mathrm{\%H}}$

$\left[\begin{array}{cc}\stackrel{\&conjugate0;}{a}& \stackrel{\&conjugate0;}{b}\\ \stackrel{\&conjugate0;}{c}& \stackrel{\&conjugate0;}{d}\end{array}\right]$

$v≔⟨1\&comma;0\&comma;-1⟩$

$v≔\left[\begin{array}{c}1\\ 0\\ \mathrm{-1}\end{array}\right]$

$A·v$

$\left[\begin{array}{c}\mathrm{-4}\\ \mathrm{-4}\end{array}\right]$

$v^{\mathrm{\%T}}$

$\left[\begin{array}{ccc}1& 0& \mathrm{-1}\end{array}\right]$

$v\&x⟨2\&comma;3\&comma;4⟩$

$\left[\begin{array}{c}3\\ \mathrm{-6}\\ 3\end{array}\right]$

$v·⟨2\&comma;3\&comma;4⟩$

$\mathrm{-2}$

$3v$

$\left[\begin{array}{c}3\\ 0\\ \mathrm{-3}\end{array}\right]$

$xA$

$\left[\begin{array}{ccc}x& 3x& 5x\\ 2x& 4x& 6x\end{array}\right]$

$x·A$

$x·\left[\begin{array}{ccc}1& 3& 5\\ 2& 4& 6\end{array}\right]$