Introductory Linear Algebra/Matrices

Motivation
One important application for matrices is solving systems of linear equations. Some of the following definitions may be viewed as 'designed for solving system of linear equations'.

Some terminologies
An $$m\times n$$ (read 'm by n') is a matrix with $$m$$ rows and $$n$$ columns, and $$m\times n$$ is the of the matrix. The rows are counted from the top, and the columns are counted from the left. If the size of a matrix is $$1\times 1$$, we simply refer to this matrix as a, and no brackets are needed in this case. The of all $$m\times n$$ matrices with  is denoted by $$M_{m\times n}(\mathbb R)$$. A letter is usually used to denote a, while  letters are used to denote its. For example, $$A=(a_{ij})_{m\times n}$$ denotes an $$m\times n$$ matrix $$A$$ with entries $$a_{ij}$$ in which $$1\le i\le\underbrace{m}_{\text{no. of rows}}$$ and $$1\le j\le \underbrace{n}_{\text{no. of columns}}$$. (We may omit the subscript specifying the size of matrix if its size is already mentioned, or its size is not important.)

In particular, if a matrix has the same number of rows and columns, then it has some nice properties. In view of the shape of such a matrix (square-like), we define such matrices as.

We will also introduce a term, namely, which will be useful in some situations.

Then, we will define some types of matrices for which the definitions are related to the.

The last terminology we mention here is, which will sometimes be used.

{{colored exercise| Choose all submatrices of $$\begin{pmatrix}3&5&7\\4&6&8\\5&7&9\end{pmatrix}$$ from the following matrices. + $$\begin{pmatrix}3&5&7\end{pmatrix}$$ - $$\begin{pmatrix}3&4&5\end{pmatrix}$$ + $$\begin{pmatrix}3\\4\\5\end{pmatrix}$$ - 2 + 7 + $$\begin{pmatrix}3&7\\5&9\end{pmatrix}$$ }}
 * type="[]"}
 * It can be obtained by removing 2nd and 3rd row of the matrix.
 * It cannot be obtained by removing some rows or columns of the matrix.
 * It can be obtained by removing 2nd and 3rd column of the matrix.
 * It cannot be obtained by removing some rows or columns of the matrix. (It is not an entry of the matrix.)
 * It can be obtained by removing 1st and 2nd column, and 2nd and 3rd row of the matrix. (Recall that $$1\times 1$$ matrix is number.)
 * Alternatively, it can also be obtained by removing 1st and 2nd row, and 1st and 3rd column of the matrix.
 * It can be obtained by removing 2nd row and 2nd column of the matrix.

Matrix operations
In this section, we will cover different matrix operations. Some operations are quite different from that in the number system, in particular, matrix multiplication.

Then, we are going to define, which is quite different from the multiplication in the number system.

On the other hand, a positive of a  is defined quite similarly to that in number system.

Then, we will discuss matrix analogs for the numbers zero and one in the number system, namely the and the, which, in the number system, are analogous to the numbers $$0$$ and $$1$$ respectively.

Then, we will introduce an operation that does not exist in the number system, namely transpose.