Matrix addition and scalar multiplication are fundamental operations in linear algebra. They form the building blocks for more complex matrix 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 scalar 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

• 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 $A + B$, is defined as the matrix obtained by adding corresponding elements: $(A + B){ij} = A{ij} + B_{ij}$ 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 $cA$, is defined as the matrix obtained by multiplying each element of A by $c$: $(cA){ij} = c(A{ij})$ for all $i$ and $j$

Examples

• If $A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}$ and $B = \begin{bmatrix} 5 & 6 \ 7 & 8 \end{bmatrix}$, then $A + B = \begin{bmatrix} 6 & 8 \ 10 & 12 \end{bmatrix}$
• If $A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}$ and $c = 3$, then $cA = \begin{bmatrix} 3 & 6 \ 9 & 12 \end{bmatrix}$

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 $A - B$, is defined as the matrix obtained by subtracting corresponding elements: $(A - B){ij} = A{ij} - B_{ij}$ for all $i$ and $j$

Examples

• If $A = \begin{bmatrix} 2 & 4 \ 6 & 8 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}$, then $A + B = \begin{bmatrix} 3 & 6 \ 9 & 12 \end{bmatrix}$
• If $A = \begin{bmatrix} 5 & 7 \ 9 & 11 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}$, then $A - B = \begin{bmatrix} 4 & 5 \ 6 & 7 \end{bmatrix}$

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 $m \times n$ matrix and $c$ is a scalar, then the scalar multiple $cA$ is an $m \times n$ matrix with elements $(cA){ij} = c(A{ij})$ for all $i$ and $j$

Examples

• If $A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}$ and $c = 2$, then $cA = \begin{bmatrix} 2 & 4 \ 6 & 8 \end{bmatrix}$
• If $A = \begin{bmatrix} 3 & 6 \ 9 & 12 \end{bmatrix}$ and $c = -1$, then $cA = \begin{bmatrix} -3 & -6 \ -9 & -12 \end{bmatrix}$

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 zero matrix, 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 $A$