Addition of two matrices having the same dimension is carried out component-wise.

Example

$\begin{bmatrix} 1 & 2 & 3\\ 4 & 5 & 6\end{bmatrix} +\begin{bmatrix} 1 & 0 & -1\\ 2 & -2 & 0\end{bmatrix}= \begin{bmatrix} 1+1 & 2+0 & 3+(-1)\\ 4+2 & 5+(-2) & 6+0\end{bmatrix}= \begin{bmatrix} 2 & 2 & 2\\ 6 & 3 & 6\end{bmatrix}$

Let $$A$$ be an $$m \times m$$ matrix. Let $$B,C$$ and $$D$$ be $$m \times n$$ matrices. Let $$E$$ be an $$n \times n$$ matrix. Then the following properties hold:

• $$B+(C+D) = (B+C)+D$$

• $$B+C = C+B$$

• $$A(B+C) = AB + AC$$

• $$(B+C)E = BE+ CE$$

Scalar multiplication

If $$\alpha$$ is a scalar and $$A$$ is a matrix, then $$\alpha A$$ denotes the matrix obtained from $$A$$ by multiplying each entry by $$\alpha$$.

Example

$(1+i)\begin{bmatrix}1 & 0 \\i & 2\end{bmatrix} =\begin{bmatrix} 1+i & 0\\ -1+i & 2 + 2i\end{bmatrix}$

Matrix subtraction is defined as follows: $$A-B= A+(-1)B$$.

Also, $$-A$$ is defined as $$(-1)A$$.

Where $$\alpha$$ and $$\beta$$ are scalars and $$A$$ and $$B$$ are $$m\times n$$ matrices $$C$$ is a $$p\times m$$ matrix, the following properties hold:

• $$\alpha(A+B)= \alpha A+\alpha B$$

• $$(\alpha+\beta)A = \alpha A + \beta A$$

• $$\alpha(\beta A) = (\alpha\beta)A$$

• $$C(\alpha A) = (\alpha C)A = \alpha(CA)$$

Exercises

1. Express each of the following in the simplest possible form.
1. $$\begin{bmatrix} 2 & 1 \\ 0 & 3 \end{bmatrix} -\begin{bmatrix} 2 & -1 \\ -1 & 2 \end{bmatrix}$$

2. $$\begin{bmatrix} 1 & 2 \\ 3 & 4 \\ 5 & 6 \end{bmatrix} + 2\begin{bmatrix} -1 & 1 \\ 1 & -1 \\ 0 & 0 \end{bmatrix}$$

2. Prove each of the listed properties.