Matrix multiplication involves summing a product. It is appropriate where you need to multiply things together and then add. As an example, multiplying the number of units by the per unit cost will give the total cost.
The units on the product are found by performing unit analysis on the matrices. The labels for the product are the labels of the rows of the first matrix and the labels of the columns of the second matrix.
Am×n × Bn×p = Cm×p
The number of columns in the first matrix must be equal to the number of rows in the second matrix. That is, the inner dimensions must be the same.
The order of the product is the number of rows in the first matrix by the number of columns in the second matrix. That is, the dimensions of the product are the outer dimensions.
Since the number of columns in the first matrix is equal to the number of rows in the second matrix, you can pair up entries.
Each element in row i from the first matrix is paired up with an element in column j from the second matrix.
The element in row i, column j, of the product is formed by multiplying these paired elements and summing them.
Each element in the product is the sum of the products of the elements from row i of the first matrix and column j of the second matrix.
There will be n products which are summed for each element in the product.
Matrix multiplication is not commutative
Do not simply multiply corresponding elements togetherMultiplication of real numbers is.
The inner dimensions may not agree if the order of the matrices is changed.
Since the order (dimensions) of the matrices don't have to be the same, there may not be corresponding elements to multiply together.
Multiply the rows of the first by the columns of the second and add.
There is no matrix division
There is no defined process for dividing a matrix by another matrix.
A matrix may be divided by a scalar.
Square matrix
Ones on the main diagonal
Zeros everywhere else
Denoted by I. If a subscript is included, it is the order of the identity matrix.
I is the multiplicative identity for matrices
Any matrix times the identity matrix is the original matrix.
Multiplication by the identity matrix is commutative, although the order of the identity may change
Identity matrix of size 2
Identity matrix of size 3
You can not change the order of a multiplication problem and expect to get the same product. AB≠BA
You must be careful when factoring common factors to make sure they are on the same side. AX+BX = (A+B)X and XA + XB = X(A+B) but AX + XB doesn't factor.
Just because a product of two matrices is the zero matrix does not mean that one of them was the zero matrix.
If A=B, then AC = BC. This property is still true, but the converse is not necessarily true. Just because AC = BC does not mean that A = B.
Because matrix multiplication is not commutative, you must be careful to either pre-multiply or post-multiply on both sides of the equation. That is, if A=B, then AC = BC or CA = CB, but AC≠CB.
You must multiply by the inverse of the matrix
Consider the product of a 2×3 matrix and a 3×4 matrix. The multiplication is defined because the inner dimensions (3) are the same. The product will be a 2×4 matrix, the outer dimensions.
Since there are three columns in the first matrix and three rows in the second matrix (the inner dimensions which must be the same), each element in the product will be the sum of three products.
To find the element in row 1, column 1 of the product, we will take row 1 from the first matrix and column 1 from the second matrix. We pair these values together, multiply the pairs of values, and then add to arrive at 25.
R1: 1 -2 3 ×C1: 1 -3 6 --------------- 1 +6 +18 = 25
Row 2, Column 3
To find the element in row 2, column 3 of the product, we will take row 2 from the first matrix and column 3 from the second matrix. We pair these values together, multiply the pairs of values, and then add to arrive at 53.
R2: 4 5 -2 ×C3: 4 7 -1 --------------- 16 +35 +2 = 53
Understanding where each number in the product comes from is helpful when you only need a specific value. You don't need to multiply completely if you only want specific elements. Just take the row from the first matrix and the column from the second matrix.
The process can be completed for the rest of the elements in the matrix.
So, the final product is