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A constant $c$ can be considered as 1 x 1 matrix $\;$ $( c )$ , it makes sense in terms of matrix inverse, matrix addition etc but the multiplication of a constant is possible to any matrix ( by multiplying all entries of matrix with it ) but if we consider constant as 1 x 1 matrix ,its multiplication with other matrices should not make sense ( as $m$ x $n$ matrix can be multiplies by $n$ x $k$ matrix only ). So why is constant multiplication defined the way it is and can we consider a constant as a 1 x 1 matrix ?

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2 Answers

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Sometimes it is convenient to treat a scalar as a $1\times 1$ matrix as we consider the base field itself as a vector space of dimension 1 and multiplication by the scalar as a linear transformation. In the same sense a scalar can be considered a vector on which we may perform such linear transformations.

When a scalar is multiplied by a matrix of arbitrary dimensions, this also treats the scalar as a linear transformation, but if we were to write this linear transformation in matrix form it would be a diagonal matrix of the correct dimensions for the multiplication to make sense. We could treat the matrices themselves as vectors in a vector space of dimension $mn$, in which case the "matrixness" of the matrix goes away and we have an $mn\times mn$ diagonal matrix. I'm either case all of the diagonal entries would be equal to the scalar.

In short, mathematical objects of the same name may wear many hats.

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Those would be two different types of products.

On one hand, if you have a $m\times n$ and a $n\times l$ matrices you define their product, which is a $m\times l$ matrix. In this sense matrices form a monoid, and as you say the product by a scalar makes sense in some cases when you consider it as the product by a $1\times 1$ matrix, because there is a bijection between scalars and $1\times 1$ matrices.

On the other hand, scalar multiplication is an example of the so called actions of a group on a set, in this case, the reals acting in the space of matrices.

In your example, the former one is a particular case of the latter one, but they are essentially different.

Edit: If you insist on using matrix multiplication to represent multiplication by scalars, all you need to do is take multiples of the identity matrix of the right dimensions.

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