Evaluate `AB={:[(3,-1),(3,5),(-2,1)]:}{:[(1,-2),(5,-3)]:}`. A is 3 by 2, and B is 2 by 2.

Explanation

The product matrix is 3 by 2, with entries `x_(ij)` calculated as follows :
`x_(il)=(3)(1)+(-1)(5)=-2`, the ''product'' of row 1 of A and column 1 of B
`x_(12)=(3)(-2)+(-1)(-3)=-3`, the ''product'' of row 1 of A and column 2 of B
`x_(21)=(3)(1)+(5)(5)=28`, the ''product'' of row 2 of A and column 1 of B
`x_(22)=(3)(-2)+(5)(-3)=-21`, the ''product'' of row 2 of A and column 2 of B
`x_(31)=(-2)(1)+(1)(5),` the ''product'' of row 3 of A and column 1 of B
`x_(32)=(-2)(-2)+(1)(-3)=1`, the ''product'' of row 3 of A and column 2 of B
`"Therefore, AB"={:[(3,-1),(3,5),(-2,1)]:}{:[(1,-2),(5,-3)]:}={:[(-2,-3),(28,-21),(3,1)]:}`. Note that BA is not defined.
Matrix calculations can be done on a graphing calculator. To define a matrix, enter 2nd MATRIX, highlight and enter EDIT, enter the number of rows followed by the number of columns, and finally enter the entries. The figures below show the result of these steps for matrices A and B of Example D.

To find the product, enter 2nd MATRIX/NAMES/[A], which returns [A] to the home screen. Also enter 2nd MATIX/NAMES/[B], which returns [B] to the screen. Hit ENTER again to get the product.

Square matrices of the same size can always be multiplied. The product of square n by n matrices is a square n by n matrix. An identity matrix `I_(n)` is a square matrix consisting of 1's down the main diagonal and 0's elsewhere. The product of n by n square matrices A and `I_(n)` is A. In other words, `I_(n)` is a multiplicative identity for n by square matrices.