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Foundations of Matrices
Introduction to the concept of matrices, their properties, and basic matrix operations.
Introduction to the concept of matrices, their properties, and basic matrix operations.
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A matrix is simply a rectangular array of numbers written in rows and columns. For example, the matrix
has 2 rows and 3 columns. We can reference a specific entry using the notation Mi,j where i is the row and j the column. For example, M1,3 is the first row and thirs column, which is π.
The order of a matrix denotes its dimensions. We say a matrix has order n×m (read "n by m") to denote that it has n rows and m columns. For example, the matrix
has order 2×3.
When matrices have the same order m×n, we can add or subtract them by simply adding or subtracting the components:
We can multiply a matrix by a scalar by simply multiplying each entry by a scalar:
If A has order m×n and B has order n×p, then the product AB is defined and will have order m×p.
Each entry of AB is the dot product of a row of A with a column of B.
For example, if we multiply a 2×2 matrix A by a 2×3 matrix B, the product will be 2×3:
In general, for matrices A and B,
Note: this does not mean AB is never equal to BA, just that it is not in general.
For all matrices A,B and C, it is true that
For all matrices A,B and C, it is true that
A matrix with all 0 entries is called the "zero" matrix, regardless of order. We denote it 0.
When you have matrices with only numbers and no parameters (eg k), you can add, multiply, and find powers of matrices using your calculator.