Matrices Formulas

Posted In: Mathematics

In mathematics, a matrix (plural: matrices) is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns.

1. An \$m\ x\ n\$ matrix A is a rectangular array of elements (numbers of functions) with m rows and n columns
\$A=[a_{ij}]=[\table a_11,a_12,…,a_{1n};a_21,a_22,…,a_{2n};…,…,…,…;a_{m1},a_{m2},…,a_{mn}]\$
2. Square matrix is a matrix of order \$nxn\$
3. A square matrix \$[a_{ij}]\$ is symmetric if \$a_{ij}=a_{ji},\$ i.e. it is symmetric around the leading diagonal. A \$4×4\$ symmetric matrix:
\$[\table 2,3,1,4;3,4,2,1;1,2,5,2;4,1,2,6]\$
4. A square matrix \$[a_{ij}]\$ is skew-symmetric if \$a_{ij}=-a_{ji}\$.
\$[\table 0,3,-1,4;-3,0,-2,1;1,2,0,-2;-4,-1,2,0]\$
5. Diagonal matrix is a square matrix with all elements zero except those on the leading diagonal. A \$4×4\$ diagonal matrix:
\$[\table 2,0,0,0;0,3,0,0;0,0,1,0;0,0,0,5]\$
6. Unit matrix is a diagonal matrix with all diagonal elements are 1. The unit matrix is denoted by I. A \$4×4\$ unit matrix:
\$I_4=[\table 1,0,0,0;0,1,0,0;0,0,1,0;0,0,0,1]\$
7. A null matrix is one whose elements are all zero. A \$4×4\$ null matrix:
\$[\table 0,0,0,0;0,0,0,0;0,0,0,0;0,0,0,0]\$