Matrix Algebra
Properties of Matrix Multiplication
Here are some standard properties of matrix multiplication: Let be an m x n matrix, and let and have sizes for which the indicated sums and products are defined.
- for any real number
- is invertible and
- for any non-zero real number
Power of a Matrix
Given an n x n matrix and a positive integer , denotes the product of copies of : . By convention, .
From the first property of the inverses, we can see that . Sometimes it is abbreviated as .
Caution: In general, because . And it equals only when .
Transpose of a Matrix
Given an m x n matrix , the transpose of is the n x m matrix, denoted by , whose columns are formed from the corresponding rows of i.e the column of is the row of .
Examples:
The following are some useful properties regarding transpose: Let A and B denote matrices having sizes for which the indicated sums and products are defined.
- for any real number
Let and be two n x n matrices. Expand .
Prove or disprove the following statement: For any invertible n x n matrices and , is also invertible and .
Prove or disprove the following statement: For any invertible n x n matrix , is also invertible and .