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Matrices

# Definitions

1. Two matrices A and B of the same order whose corresponding entries are equivalent are considered equal. That is: aij=bij i=1,...,n ; j=1,2,...,m.
2. A square matrix is a matrix which has the same number of rows and columns.
3. The main diagonal of a matrix is formed by the elements of a matrix starting in the upper left corner and proceeding down and to the right (aii).
4. A matrix with all-zero entries below the top-left-to-lower-right diagonal is called upper triangular.
5. A matrix with all-zero entries over the top-left-to-lower-right diagonal is called lower triangular.
6. A matrix with non-zero entries only on the diagonal is called diagonal.
7. A diagonal matrix whose non-zero entries are all 1's is called an identity matrix.
The 3 × 3 identity is denoted by I3 (pronounced as "eye-three" or "eye-sub-three"); similarly, the 4 × 4 identity is I4 and the 5 × 5 identity matrix is I5.
8. The transpose of a Matrix A is a matrix which is formed by turning all the rows of the given matrix into columns and vice-versa. The transpose of matrix A is written AT.
 Matrix with only one row Matrix with only one column Upper triangular Lower diagonal Diagonal $\fs2A=\left(\begin{matrix}1&-1&4\end{matrix}\right)$ $\fs2A=\left(\begin{matrix}-3\\-1\\2\end{matrix}\right)$ $A=\left(\begin{matrix}2&-1&0\\0&12&1\\0&0&5\end{matrix}\right)$ $A=\left(\begin{matrix}2&0&0\\3&-2&0\\2&-1&5\end{matrix}\right)$ $A=\left(\begin{matrix}2&0&0\\0&1&0\\0&0&5\end{matrix}\right)$