Multiplying Matrices

Scalar and matrix multiplication will be discussed since they involve two different concepts as well as different rules.

Scalar Multiplication

Definition:

In scalar multiplication of a matrix the scalar is multiplied by each entry of the matrix.

Let c = scalar and . The product of the scalar and a matrix is:

Multiplication by Another Matrix

Definition:

For given matrices X and Y, where X is a (m x t) matrix and Y is a (t x n) matrix, the product of XY is an m x n matrix whose entries are determined as follows. To find the entry of row m and column n of matrix XY, find row m of matrix X and column n of matrix Y and multiply the corresponding entries and add all the products together.

A convenient way to determine if two matrices can be multiplied is to write down the size of the matrices in the order of which they will be multiplied.

The matrices can be multiplied if the two inside terms are equal, i.e., the mumber of columns in X equals the number of rows in Y. The two outside terms determine the size of the resulting matrix. When describing two matrices being multiplied one should note the order.

If two matrices, X and Y, are multiplied as follows:

XY

then X is pre-multiplied by Y and Y is post-multiplied by X.

For example, consider the following matrices

The first thing that must be done is to verify if the two matrices can be multiplied. By definition the number of columns of the first matrix must equal the number of rows of the second matrix. Matrix X has three columns and matrix Y has three rows. The resulting matrix will be a (2 x 3) matrix.

In other words, the order is (2 x 3) since X has two rows and Y has three columns, respectively.

Next we must determine the entries of matrix XY. For example to determine the entry in the first row and first column of matrix XY (i.e., a₁₁), we usethe first row of X and the first column of Y, multiply the corresponding entries, and add them together.

The entry of row one and column two of XY (i.e., a₁₂) is computed as follows

After full multiplication the following matrix results.

Properties of Multiplication

• Associative Law.   Unlike conventional multiplication matrix A must be multiplied from left to right. Although, it doesn't matter how the multiplication is grouped.

• Distributive Law.   The distributive property applies for multiplication. Matrix A can be multiplied by quantity B + C or A can be distributed and the two products can be added.

• cA = Ac, where c is a scalar   Multiplication of a matrix by a scalar is simple because each entry of the matrix is multiplied by the same value.

• A(cB) = c(AB) = (cA)B, where c is a scalar.   As you can see the order of the matrices is still important but the order in which the constant is multiplied is not important.

Examples:

Determine 2A , A + B, AB, and BA for the following matrices:

A + B is undefined for this problem. The order of A does not equal B. A is a matrix of order (3 x 3) and B is a (2 x 3) matrix.

AB is undefined for this problem. The two inner indices are not equal, i.e., (3 x 3) * (2 x 3).

To multiply BA one must:

1) Verify they can be multiplied. We can see the inner terms are equal and will form a matrix of order (2 x 3).

2) Next compute the entries

3) Assemble the matrix