### Systems of linear equations and matrices: Matrices

### Simple matrix operations

Under certain conditions we can calculate with matrices. Here we discuss addition and scalar multiplication. We also look at reflection about the main diagonal.

Addition of matrices

If \(A\) and \(B\) are matrices of the same size, then the **sum matrix** \(A+B\) is the matrix that you will get by adding corresponding elements. Just as for numbers, the operation is called **addition**.

Addition of matrices satisfies the following two properties, where \(A=(a_{ij})\), \(B=(b_{ij})\) and \(C=(c_{ij})\) are three \((m\times n)\)-matrices:

\[

\begin{array}{ll}

A+B=B+A &\phantom{xxx} \color{blue}{\text{commutativity}} \\

(A+B)+C=A+(B+C) &\phantom{xxx} \color{blue}{\text{associativity}}

\end{array}

\]

Written out in coordinates this definition reads as follows:

Let \(A\) and \(B\) both be \((m\times n)\)-matrices with elements \(a_{ij}\) and \(b_{ij}\), respectively. For \( 1\leq i\leq m\) and \(1\leq j\leq n\) define \[

c_{ij}=a_{ij}+b_{ij}

\] The \((m\times n)\)-matrix \(C\) with elements \(c_{ij}\) is the **sum** of the matrices \(A\) and \(B\).

The associativity enables us to talk about \( A+B+C\) without specifying how we determine the matrix: as \((A+B)+C\) or as \(A+(B+C)\). The result is the same either way.

Scalar multiplication of a matrix If \(A\) is a matrix and \(\lambda\) a number, then \(\lambda\cdot A\) or simply #\lambda A#, the matrix you get by multiplying all the elements of \(A\) by \(\lambda\). We call this operation the **scalar multiplication** of the scalar #\lambda# by the matrix #A# and the result the **scalar product**.

If #\lambda = -1#, we often write #-A# rather than #-1 A#. This matrix is called the **opposite** **matrix** of #A#.

For scalar multiplication calculation rules are as shown below, where \(A\) and \(B\) are matrices of equal size, and \(\lambda\) and \(\mu\) are scalars:

\[

\begin{array}{rl}

1\,A\!\!\! & =A \\

(\lambda+\mu)\,A\!\!\! &= \lambda\, A+\mu\, A \\

\lambda\,(A+B)\!\!\! &= \lambda A+\lambda B \\

\lambda(\mu\, A)\!\!\! &= (\lambda\, \mu)\, A

\end{array}

\]

The transposed matrix The **transposed** matrix of \(A\), denoted as \(A^{\top}\), is the matrix that you get when you reflect \(A\) about its main diagonal. If \(A\) is an \((m\times n)\)-matrix, then \(A^{\top}\) is an \((n\times m)\)-matrix.

For matrices \(A\) and \(B\) of equal size and each scalar \(\lambda\) we have \[

\begin{array}{rl}

(A+B)^{\top}\!\!\! & = A^{\top}+B^{\top} \\

(\lambda A)^{\top}\!\!\! & = \lambda A^{\top} \\

(A^{\top})^{\top}\!\!\! & =A

\end{array}

\]

Symmetric and anti-symmetric matrices A **symmetric matrix** is a square matrix that is equal to its transpose. An **anti-symmetric matrix** is a square matrix that is opposite to its transpose.

In other words, a matrix \(A\) is\[\begin{array}{rrcl}\text{symmetric if }&A^{\top}&=&A\\ \text{anti-symmetric if }&A^{\top}&=&-A\end{array}\]

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