Matrix Multiplication: In algebra there are two types Matrix multiplication: Multiplication of a matrix by a number and multiplication of a matrix by another matrix.
Scalar Multiplication: When we multiply a matrix with a number it is called scalar multiplication in which every element of a matrix is multiplied by the same number.
For example: If we multiply a matrix A with 5 then every element of the matrix would be multiplied with 5.
Multiplication of a Matrix with another Matrix:
The matrix product AB is defined only when the number of columns in A is equal to the number of rows in B. Similarly, the matrix product BA is defined only when the number of columns in B is equal to the number of rows in A. This type of multiplication is also known as dot product.
Suppose A is a m*n matrix and B is a n*o matrix, then matrix product AB results into a matrix C with dimensions of m*o i.e. C would be a matrix with m rows and o columns. Each element of the C matrix would be computed as per the following formula.
Cmo = ∑n Amn*Bno
The multiplication operation in the above matrix happened as per below equations:
- C11 = 0*6 + 1*8 +2*10 = 0 + 8 + 20 = 28
- C12 = 0*7 + 1*9 +2*11 = 0 + 9 + 22 = 31
- C21 = 3*6 + 4*8 +5*10 = = 18 + 32 + 50 = 100
- C22 = 3*7 + 4*9 +5*11 = 21 + 36 +55 = 112
In some cases, matrix multiplication is defined for AB but not for BA and vice versa. However, even when matrix multiplication is possible in both directions, results may be different. That is AB is not always equal to BA.
Because order is important, matrix algebra jargon has evolved to clearly indicate the order in which matrices are multiplied.
- To describe the matrix product AB, we can say A is post-multiplied by B; or we can say that B is pre-multiplied by A.
- Similarly, to describe the matrix product BA, we can say B is post-multiplied by A; or we can say that A is pre-multiplied by B.
The bottom line: when you multiply two matrices, order matters.
The identity matrix is an n x n diagonal matrix with 1’s in the diagonal and zeros everywhere else. The identity matrix is denoted by I or In. Two identity matrices appear below.
The identity matrix has a unique talent. Any matrix that can be pre-multiplied or post-multiplied by I remains the same; that is:
AI = IA = A