More about this Matrix Multiplication Calculator

Matrices appear frequently in linear algebra because of their intimate connection with linear functions. But besides that link, matrices are objects that behave a lot like numbers. Indeed, you can add, subtract and multiply matrices, provided that the dimensions are compatible.

For example, in order to add two matrices you need to have that they have the same dimensions. The same requirement is needed for when you want to subtract matrices.

How do you multiply matrices?

Multiplication of matrices poses a different challenge, as its definition is less intuitive as the way we add and subtract matrices. Also, the suitable dimensions for the multiplication do not require that the matrices have the same dimensions but yet a different condition.

So, let's start with that: in order to be able to multiply matrices, the number of columns of the first matrix needs to coincide with the number of rows of the second matrix.

This actually means that you can have two matrices of different shapes that can be multiplied. For example, a 2x4 matrix can be multiplied by a 4x4 matrix. Or a 3x3 matrix can be multiplied by a 3x6 matrix.

Now, how do you define the multiplication between two matrices? You do the definition componentwise as follows: Assume that \(A\) is a \(m \times n\) matrix and \(B\) is a \(n \times p\) matrix

\[ (A B)_{ij} = \sum_{k=1}^n A_{ik} B_{kj}\]

Often times this formula can be hard to digest, but the best way to do it is to think of it like this: the element of the product matrix that is in row i and column j is computed by computing the dot product between the i-th row of the first matrix and the j-th column of the second matrix.

What is the identity matrix property of matrix multiplication?

The identity matrix is very special in terms of matrix multiplication. Indeed, a matrix A does not change at all when multiplied by the identity matrix (provided that the dimensions are valid to conduct the multiplication)

Is this a Matrix multiplication calculator with steps?

Yes it is. All you need to do is to provide the matrices you want to multiply, and the calculator will do the rest. The calculator starts with two empty 2x2 matrices. So, you may need to adjust the dimensions of the matrices to type in the matrices you need.

Is this a 3 matrix multiplication calculator?

Not directly. This calculator will compute the product of two matrices. If you want to multiply three functions, then you need to first compute the multiplication of the first two, and then the result multiply it by the third.