Why do we learn matrix in math

Matrix calculation - basics

Special matrices

In the following, some matrices are named which differ from other matrices due to their special shape.

Square matrices

Matrices that have the same number of rows and columns (\ (m = n \)) are called quadratic. Well-known representatives of this genus are the 2x2 and 3x3 matrices, which are often found in schools and studies.

\ (A = \ begin {pmatrix} {\ color {red} a_ {11}} & a_ {12} & a_ {13} \ a_ {21} & {\ color {red} a_ {22}} & a_ {23} \ a_ {31} & a_ {32} & {\ color {red} a_ {33}} \ end {pmatrix} \)

The elements of a square matrix, for which \ (i = j \) applies, form the so-called. Main diagonal the matrix.

Zero matrix

If all elements of a matrix are equal to zero, it is called a zero matrix.

\ (A = \ begin {pmatrix} 0 & 0 \ 0 & 0 \ end {pmatrix} \)

A 2x2 zero matrix is ​​used as an example.

Identity matrix

A matrix in which the elements of the main diagonal are equal to one and all other elements are equal to zero is called an identity matrix.

\ (A = \ begin {pmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \ end {pmatrix} \)

Diagonal matrix

A matrix in which all elements - except the elements of the main diagonal - are equal to zero is called a diagonal matrix.

\ (A = \ begin {pmatrix} 3 & 0 & 0 \ 0 & -4 & 0 \ 0 & 0 & 2 \ end {pmatrix} \)

Note: The identity matrix (elements of the main diagonal equal to one) and the zero matrix (elements of the main diagonal equal to zero) are special diagonal matrices.

Upper triangular matrix

If all elements below the main diagonal are equal to zero, the matrix is ​​called the "upper triangular matrix".

\ (A = \ begin {pmatrix} {\ color {red} 3} & {\ color {red} 4} & {\ color {red} 1} \ 0 & {\ color {red} -5} & { \ color {red} 4} \ 0 & 0 & {\ color {red} 4} \ end {pmatrix} \)

Lower triangular matrix

If all elements above the main diagonal are equal to zero, the matrix is ​​called the "lower triangular matrix".

\ (A = \ begin {pmatrix} {\ color {red} 3} & 0 & 0 \ {\ color {red} 1} & {\ color {red} -2} & 0 \ {\ color {red } 5} & {\ color {red} 5} & {\ color {red} 4} \ end {pmatrix} \)

More matrices ...

We have written a separate article for each of the following matrices

As you can see, the topic of matrix calculations is a relatively large sub-area of ​​mathematics. This is mainly due to the great importance of matrices in practice.