Matrix addition

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:

• \(\alpha(A+B)= \alpha A+\alpha B\)

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

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

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

Quick Quiz

Exercises

1. Express each of the following in the simplest possible form.

   1. \(\begin{bmatrix} 2 & 1 \\ 0 & 3 \end{bmatrix} -\begin{bmatrix} 2 & -1 \\ -1 & 2 \end{bmatrix}\)  

      \(\begin{bmatrix} 0 & 2 \\ 1 & 1\end{bmatrix}\)

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

      \(\begin{bmatrix} -1 & 4 \\ 5 & 2 \\ 5 & 6 \end{bmatrix}\)

2. Prove each of the listed properties.