That is, the matrix kA is obtained by multiplying each entry of A by k. If A = is a matrix and k is a scalar, then A matrix can be multiplied by a scalar as follows. Since the equation A + B = C is equivalent to B = C − A, employing matrix subtraction above would yield the same result: This example motivates the definition of matrix subtraction: If A and B are matrices of the same size, then the entries of A − B are found by simply subracting the entries of B from the corresponding entries of A. Since two matrices are equal if and only if they are of the same size and their corresponding entries are equal, this last equation implies Then the matrix equation A + B = C becomes

This is the matrix analog of the statement a + 0 = 0 + a = a, which expresses the fact that the number 0 is the additive identity in the set of real numbers.Įxample 3: Find the matrix B such that A + B = C, where Since addition of real numbers is commutative, it follows that addition of matrices (when it is defined) is also commutative that is, for any matrices A and B of the same size, A + B will always equal B + A.Įxample 2: If any matrix A is added to the zero matrix of the same size, the result is clearly equal to A:

Since only matrices of the same size can be added, only the sum F + H is defined ( G cannot be added to either F or H). Which two can be added? What is their sum? Thus, to find the entries of A + B, simply add the corresponding entries of A and B.Įxample 1: Consider the following matrices: (This is similar to the restriction on adding vectors, namely, only vectors from the same space R ncan be added you cannot add a 2‐vector to a 3‐vector, for example.) If A = and B = are both m x n matrices, then their sum, C = A + B, is also an m x n matrix, and its entries are given by the formula

If A and B are matrices of the same size, then they can be added.

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