The Matrix Multiplication Calculator performs the following matrix operation:
Matrices consist of rows and columns, where given a matrix `A`, the position in `A` in vCalc is denoted `A_(ij)` where the `1^(st)` subscript indicates the row of the matrix and the `2^(nd)` subscript indicates the column of the matrix. We refer to `A_(ij)` as the `(i, j)"th"` element of the matrix `A`. An arbitrary matrix has its size denoted as `mtimesn`, where `m` refers to the number of rows in a given matrix and `n` refers to the number of columns in a given matrix.
If `m=n` then the matrix is referred to as a square matrix. The elements of the matrix `A_(11), A_(22), ..., A_(text(nn))` is commonly referred to as the main diagonal of the square matrix.
Let `A` be a matrix and `c` be an arbitrary scalar number; scalar multiplication of `A` by `c` is "the matrix obtained by multiplying every element of `A` by `c`. The matrix `cA` will be the same size as `A`" (Williams, 37).
Multiplying matrices is done by multiplying the rows of the first matrix with the columns of the second matrix in a systematic manner. In order for us to be able to multiply two matrices together, the number of columns in `A` has to be equal to the number of rows in `B`. Otherwise, the product `AB` of two matrices does not exist.
The matrix multiplication rule is as follows:
"Interpret the first matrix of a product in terms of its rows and the second in terms of its columns. Multiply rows times columns by multiplying corresponding elements and adding" (Williams, 37).
Consider the following matrices `A` and `B`:
`A= [(3, 1, 2), (4, 1, 5)], B=[(7, 2), (6, 3), (5, 1)]`
Since `A` has three columns and `B` has three rows, we know we can multiply these matrices to get a new matrix. We use the `AB` multiplication rule to get
`AB= [( (3*7)+(1*6)+(2*5) , (3*2)+(1*3)+(2*1)), ((4*7)+(1*6)+(5*5) , (4*2)+(1*3)+(5*1))]`
`AB=[(37, 11), (59, 25)]`
Now let's note an example from Williams on page 39:
"Consider the following matrices `A` and `B`:
`A= [(3, 1, 2), (4, 1, 5)], B=[(7, 2), (6, 3)]`
Let us attempt to compute `AB` using the matrix multiplication rule. We get
`AB= [(3, 1, 2), (4, 1, 5)]*[(7, 2), (6, 3)]= [([(3, 1, 2)]*[(7), (6)],[(3, 1, 2)]*[(2), (3)]),( [(4, 1, 5)]*[(7), (6)], [(4, 1, 5)]*[(2), (3)]) ]`
If we try to compute `[(3, 1, 2)]*[(7), (6)] `, the elements do not match, and the product does not exist. The same shortcoming applies to all the other elements of `AB`. We say that the product `AB` does not exist."
The following properties of matrix multiplication are important to know: 1) Matrix Multiplication is not commutative 2) If `A` is an `m times r` matrix and `B` is an `r times n` matrix, then `AB` will be an `mtimesn` matrix.
Williams, Gareth. Linear Algebra With Applications. Boston: Jones and Bartlett, 2011. Print.