Many computer applications require large amounts of numerical data laid out in two-dimensional arrays. Computer graphics, mathematical graph theory, and probability/statistics are just some of these applications.

A matrix is a two-dimensional structure of numerical values. Below is a 4 × 3 matrix (4 rows, 3 columns) with row and column numbers shown in black, and matrix elements shown in red. Note:

• The row and column numbers begin with zero
• If this matrix is named A, its elements are referred to by a_ij, where i and j are the element's row and column numbers, respectively.

  Thus the upper left element of A is referred to as a₀₀, and the lower right element as a₃₂.

The transpose of an m × n matrix A is an n × m matrix whose i^th column has the same values as A's i^th row.

More formally, if B is the transpose of A, then for all elements b_ij of B, b_ij = a_ji.

Also, if B is the transpose of A, then A is the transpose of B.

For example, the matrices below are transposes of each other.

   

An m × m (i.e. square) matrix is an identity matrix if all of its elements are zeros (0), except for those along the diagonal, which all are ones (1).

More formally, if A is an identity matrix, then all elements a_ii equal 1, while the rest equal 0.

Shown below is a 4 × 4 identity matrix. An identity matrix is so-called because when it is multiplied by a suitable m × k matrix M (see matrix multiplication below), the result is just M.

Two matrices are equal if they have the same dimensions and their elements are pair-wise equal.

That is, matrix A is equal to B if A and B have the same dimensions and a_ij = b_ij for all i and j within range.

Examples are shown in the menu to the left. Note that the examples are snapshots of a matrix manipulation tool you will use in a lab exercise and programming assignment.

In this example, the matrices are not equal because the elements in the bottom right location are not equal.

In this example, all elements are equal so the matrices are equal.

In this example, the matrices' dimensions are not the same, so the matrices are not equal.

To add two matrices, they must have the same dimensions, i.e. the same number of rows and columns.

The result of adding matrices A and B is a matrix C of the same dimensions, where c_ij = a_ij + b_ij.

For example:

To multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. Matrix multiplication is not commutative.

For example, consider these matrices:

The product of Matrix 1 and Matrix 2 is defined, but the product of Matrix 2 and Matrix 1 is not.

The result of multiplying an m×n matrix A and an n×p matrix B is an m×p matrix C where c_ij is the dot product of the i^th row of A and the j^th column of B.

For example,

• The third row (index 2) of Matrix 1 contains the values  6   5   1 
• The first column (index 0) of Matrix 2 contains the values  2   4   3 
• The dot product of  6   5   1  and  2   4   3  =
   6×2 + 5×4 + 1×3  =
   12 + 20 + 3  =
   35 
• So the first column of the third row of Result = 35

Computing all possible dot products yields:

If Matrix 1 is an m×n matrix and Matrix 2 is an n×n identity matrix, then the result of multiplying Matrix 1 by Matrix 2 is Matrix 1.

See the menu at left for examples.

This section describes simple actions for automatically setting the elements of matrices.

We can fill a matrix "row-wise" by:

• Setting the upper left element to 1 and incrementing successive elements of the first row by 1 from left to right
• When a row is filled, move to the next row and repeat, starting with the last element value plus 1

Here is the result of filling a 4×3 matrix row-wise:

We can fill a matrix "column-wise" by:

• Setting the upper left element to 1 and incrementing successive elements of the first column by 1 from top to bottom
• When a column is filled, move to the next column and repeat, starting with the last element value plus 1

Here is the result of filling a 4×3 matrix column-wise: