Linear Algebra/Addition, Multiplication, and Transpose

Addition and subtraction


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.




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



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.



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

  3. 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