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Example:Verify Property of Transpose

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Matrix (mathematics)


Specific elements of a matrix are often denoted by a variable with two subscripts. For instance,a2,1 represents the element at the second row and first column of a matrix A.

In mathematics, a matrix (plural matrices, or less commonly matrixes) is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements is




Matrix addition


In mathematics, matrix addition is the operation of adding two matrices by adding the corresponding entries together. However, there is another operation which could also be considered as a kind of addition for matrices.

The usual matrix addition is defined for two matrices of the same dimensions. The sum of two m-by-n matrices A and B, denoted by A + B, is again an m-by-n matrix computed by adding corresponding elements. For example:

(Our solved example in mathguru.com uses this concept)

We can also subtract one matrix from another, as long as they have the same dimensions. A - B is computed by subtracting corresponding elements of A and B, and has the same dimensions as A and B. For example:






In linear algebra, the transpose of a matrix A is another matrix AT (also written A′, Atr or tA) created by any one of the following equivalent actions:

1.  reflect A over its main diagonal (which runs top-left to bottom-right) to obtain AT

2.  write the rows of A as the columns of AT

3.  write the columns of A as the rows of AT

4.  visually rotate A 90 degrees clockwise, and mirror the image in a vertical line to obtain AT

Formally, the (i,j) element of AT is the (j,i) element of A.

[AT]ij = [A]ji

(Our solved example in mathguru.com uses this concept)

If A is an m × n matrix then AT is a n × m matrix. The transpose of a scalar is the same scalar.









The above explanation is copied from Wikipedia, the free encyclopedia and is remixed as allowed under the Creative Commons Attribution- ShareAlike 3.0 Unported License.