Linear Algebra/Addition, Multiplication, and Transpose

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Addition and subtraction[edit]

Two matrices can only be added or subtracted if they have the same size. Matrix addition and subtraction are done entry-wise, which means that each entry in A+B is the sum of the corresponding entries in A and B.

Here is an example of matrix addition

And an example of subtraction

Remember you can not add or subtract two matrices of different sizes.

The following rules applies to sums and scalar multiples of matrices.
Let be matrices of the same size, and let be scalars.

Multiplication[edit]

What is matrix multiplication? You can multiply two matrices if, and only if, the number of columns in the first matrix equals the number of rows in the second matrix.

Otherwise, the product of two matrices is undefined. The product matrix's dimensions are

In above multiplication, the matrices cannot be multiplied since the number of columns in the 1st one, matrix is not equals the number of rows in the 2nd, matrix . The Dimensions of the product matrix. Rows of 1st matrix × Columns of 2nd

Powers[edit]

If is an matrix and if is a positive integer, then denotes the product of copies of

If is non-zero and if is in , then is the result of left-multiplying by repeatedly times. If , then should be itself. Thus is interpreted as the identity matrix.

Transpose[edit]

Given the matrix , the transpose of is the , denoted , whose columns are formed from the corresponding rows of .

For example

The following rules applied when working with transposing

  1. For any scalar ,

The 4th rule can be generalize to products of more than two factors, as "The transpose of a product of matrices equals the product of their transposes in the reverse order." Meaning