and are fundamental operations in linear algebra. They form the building blocks for more complex manipulations, allowing us to combine and scale matrices in various ways. These operations are essential for solving systems of equations and transforming vectors.
Understanding these operations is crucial for grasping more advanced concepts in linear algebra. Matrix addition lets us combine information from different sources, while multiplication allows us to scale or shrink matrices uniformly. These tools are vital for working with linear transformations and solving real-world problems.
Matrix addition and scalar multiplication
Definition and notation
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Matrix addition is the process of adding two matrices of the same size by adding corresponding elements in each matrix
The sum of two matrices A and B, denoted [A + B](https://www.fiveableKeyTerm:a_+_b), is defined as the matrix obtained by adding corresponding elements: (A+B)ij=Aij+Bij for all i and j
Scalar multiplication is the process of multiplying a matrix by a scalar (a real number) by multiplying each element of the matrix by the scalar
The scalar multiple of a matrix A by a scalar c, denoted [cA](https://www.fiveableKeyTerm:ca), is defined as the matrix obtained by multiplying each element of A by c: (cA)ij=c(Aij) for all i and j
Examples
If A=[1324] and B=[5768], then A+B=[610812]
If A=[1324] and c=3, then cA=[39612]
Matrix addition and subtraction
Computing the sum and difference of matrices
To add matrices, ensure that the matrices have the same dimensions (i.e., the same number of rows and columns)
Add the corresponding elements of the matrices to obtain the sum matrix
To subtract matrices, ensure that the matrices have the same dimensions
Subtract the corresponding elements of the second matrix from the first matrix to obtain the difference matrix
The difference of two matrices A and B, denoted [A - B](https://www.fiveableKeyTerm:a_-_b), is defined as the matrix obtained by subtracting corresponding elements: (A−B)ij=Aij−Bij for all i and j
Examples
If A=[2648] and B=[1324], then A+B=[39612]
If A=[59711] and B=[1324], then A−B=[4657]
Scalar multiplication of matrices
Calculating the scalar multiple of a matrix
To calculate the scalar multiple of a matrix, multiply each element of the matrix by the given scalar
The resulting matrix will have the same dimensions as the original matrix
If A is an m×n matrix and c is a scalar, then the scalar multiple cA is an m×n matrix with elements (cA)ij=c(Aij) for all i and j
Examples
If A=[1324] and c=2, then cA=[2648]
If A=[39612] and c=−1, then cA=[−3−9−6−12]
Properties of matrix operations
Matrix addition properties
Matrix addition is commutative: A+B=B+A for all matrices A and B of the same size
Matrix addition is associative: (A+B)+C=A+(B+C) for all matrices A, B, and C of the same size
The , denoted 0, is the identity element for matrix addition: A+0=A for any matrix A
For any matrix A, there exists an additive inverse −A such that A+(−A)=0
Scalar multiplication properties
Scalar multiplication is distributive over matrix addition: c(A+B)=cA+cB for any scalar c and matrices A and B of the same size
Scalar multiplication is compatible with scalar addition: (c+d)A=cA+dA for any scalars c and d and matrix A
Scalar multiplication is compatible with scalar multiplication: c(dA)=(cd)A for any scalars c and d and matrix A
The scalar 1 is the identity element for scalar multiplication: 1A=A for any matrix A