LA - Difficult knowledge about matrices

MATRICES

Zero matrix: all entries in the matrix are zero

Main diagonal: all entries on the main diagonal, starting from $a_{11}$

Square matrix of order $n$: $A_{n*n}$

Diagonal matrix: square matrix with all entries except the main diagonal are zero

Triangular matrix:

We have two kinds, this is upper and the reverse is lower

Identity matrix: square matrix with 1s on the main diagonal and the remaing are 0s.

Tranpose of $A$: $A^T$: when we put all the entries on the rows to the columns and all on the columns to the rows.

Elementary Row Operations on matrix $A$:

Interchange 2 rows $r_i \leftrightarrow r_j$
Multiply a row through by a nonzero constant: $r_i \to \lambda r_i$ $(\lambda \ne 0)$
Add a constanst times one row to another: $r_i \to r_j + \lambda . r_j$

Leading entry: first non-zero element of each row

Row echelon form: <in Vietnamese> phần tử của ma trận hàng dưới sẽ thụt vô so với hàng trên

Rank of the matrix: the number of non-zero rows of matrix, denoted by $r(A)$

Two equal matrices when they have the same size and their corresponding entries are equal.

The product $\alpha A$ is obtained by multiplying each entry of the matrix $A$ by $\alpha$. (scalar multiple)

$A_(m*n) * B_(n*p) = C_(m*p)$

<Vietnamese> Lấy hàng nhân cột, lấy tổng các tích.

Some properties:

$AI = A, \forall A$

$(AB)^T = B^T + A^T$

$AB \# BA$

Elementary matrix: a matrix can be obtained from an identity matrix $I$ by performing a single elementary row operation.

If elementary matrix $E$ by performing a row operation on $I_m$ then $EA$ is the matrix resulted from performing the same row operation on $A$

If $A$ is a square matrix, we can find a matrix $B$ such that

$AB = BA = I$

$\to A$ is invertible and $B$ is an inverse of $A$, if no $B$ is found then $A$ is said to be singular