Matrix Addition and Multiplication »

Square matrix:

If a matrix A has n rows and n columns then it can be said that it's a square matrix.

Example 1:

$ A = \left( {\begin{array}{*{20}{c}} 1&2&3\\ 4&5&6\\ 7&8&9 \end{array}} \right) $

Diagonal matrix

A diagonal matrix is a square matrix with all non-diagonal elements being 0. The diagonal matrix is completely denoted by the diagonal elements.

Example 2:

$ A = \left( {\begin{array}{*{20}{c}} 1&0&0\\ 0&5&0\\ 0&0&9 \end{array}} \right) $

The matrix is denoted by the diagonal $(1 , 5 , 9)$

Row matrix

A matrix with one row is called the row matrix

Example 3:

$ A = \left[ {\begin{array}{*{20}{c}} 1&2&3 \end{array}} \right] $

Column matrix

A matrix with one column is called the column matrix

Example 4:

$ A = \left[ {\begin{array}{*{20}{c}} 1\\ 4\\ 7 \end{array}} \right] $

Matrices of the same kind

Matrix $A$ and $B$ are of the same kind if $A$ has as many rows and as many columns as $B$

The transpose of a matrix

The $n \times m$ matrix $A_T$ is the transpose of the $m \times n$ matrix $A$ if and only if the ith row of $A$ is equal to the ith column of $A_T$ for $(i = 1,2,3,..n)$.

Example 5:

$ A = \left[ {\begin{array}{*{20}{l}} {\color{red}{1}}&{\color{red}{2}}&{\color{red}{3}}\\ 4&5&6 \end{array}} \right] \to {A^T} = \left[ {\begin{array}{*{20}{l}} {\color{red}{1}}&4\\ {\color{red}{2}}&5\\ {\color{red}{3}}&6 \end{array}} \right] $

0-matrix

When all the elements of a matrix $A$ are 0, we call it the 0-matrix.

Example 6:

$ A = \left[ {\begin{array}{*{20}{c}} 0&0&0\\ 0&0&0\\ 0&0&0 \end{array}} \right] $

An identity matrix $I$

An identity matrix $I$ is a diagonal matrix with all diagonal element equal to 1

Example 7:

$ A = \left[ {\begin{array}{*{20}{c}} 1&0&0\\ 0&1&0\\ 0&0&1 \end{array}} \right] $

The opposite matrix of a matrix

If the sign of all the elements of a matrix $A$ are changed, that matrix is the opposite matrix $-A$.

Example 8:

$ A = \left( {\begin{array}{*{20}{c}} 1&{ - 2}&3\\ { - 4}&{ - 5}&{ - 6}\\ { - 7}&8&9 \end{array}} \right) \to A' = \left( {\begin{array}{*{20}{c}} { - 1}&2&{ - 3}\\ 4&5&6\\ 7&{ - 8}&{ - 9} \end{array}} \right) $

A symmetric matrix

A square matrix is called symmetric if it is equal to its transpose.

Example 9:

$ A = \left( {\begin{array}{*{20}{c}} 1&2&3\\ 2&5&6\\ 3&6&9 \end{array}} \right) $