# Nakafa Learning Content > For AI agents: use [llms.txt](https://nakafa.com/llms.txt) for the site index. Markdown versions are available by appending `.md` to content URLs or sending `Accept: text/markdown`. URL: https://nakafa.com/en/subjects/mathematics/matrix/matrix-types Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/material/lesson/mathematics/matrix/matrix-types/en.mdx Learn the main matrix types: row, column, square, triangular, diagonal, identity, zero, and symmetric matrices, with definitions and examples. --- ## Row Matrix A row matrix is a matrix that consists of only one row. The order of a row matrix is $$1 \times n$$, where $$n$$ is the number of columns. Visible text: The order of a row matrix is , where is the number of columns. Its general form is: ```math A_{1 \times n} = \begin{bmatrix} a_{11} & a_{12} & \dots & a_{1n} \end{bmatrix} ``` Example: ```math P = \begin{bmatrix} 3 & -1 & 0 & 7 \end{bmatrix} ``` Matrix $$P$$ is a row matrix of order $$1 \times 4$$. Visible text: Matrix is a row matrix of order . ## Column Matrix A column matrix is a matrix that consists of only one column. The order of a column matrix is $$m \times 1$$, where $$m$$ is the number of rows. Visible text: The order of a column matrix is , where is the number of rows. Its general form is: ```math B_{m \times 1} = \begin{bmatrix} a_{11} \\ a_{21} \\ \vdots \\ a_{m1} \end{bmatrix} ``` Example: ```math Q = \begin{bmatrix} 2 \\ 5 \\ -4 \end{bmatrix} ``` Matrix $$Q$$ is a column matrix of order $$3 \times 1$$. Visible text: Matrix is a column matrix of order . ## Square Matrix A square matrix is a matrix that has the same number of rows and columns. If $$\text{number of rows} = \text{number of columns} = n$$, then the matrix is of order $$n \times n$$. Visible text: If , then the matrix is of order . Its general form is: ```math A_{n \times n} = \begin{bmatrix} a_{11} & a_{12} & \dots & a_{1n} \\ a_{21} & a_{22} & \dots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \dots & a_{nn} \end{bmatrix} ``` In a square matrix, there are: 1. **Main Diagonal** (or Principal Diagonal): The elements $$a_{11}, a_{22}, \dots, a_{nn}$$ (i.e., $$a_{ij}$$ where $$i=j$$). 2. **Anti-diagonal** (or Counter-diagonal): The elements $$a_{1n}, a_{2,n-1}, \dots, a_{n1}$$ (i.e., $$a_{ij}$$ where $$i+j=n+1$$). Visible text: 1. **Main Diagonal** (or Principal Diagonal): The elements (i.e., where ). 2. **Anti-diagonal** (or Counter-diagonal): The elements (i.e., where ). Example: ```math M = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} ``` Matrix $$M$$ is a square matrix of order $$3 \times 3$$. Its main diagonal elements are $$1, 5, 9$$. Its anti-diagonal elements are $$3, 5, 7$$. Visible text: Matrix is a square matrix of order . Its main diagonal elements are . Its anti-diagonal elements are . ## Rectangular Matrix A rectangular matrix is a matrix where the number of rows and columns are not equal ($$m \neq n$$). Visible text: A rectangular matrix is a matrix where the number of rows and columns are not equal (). General example: ```math C = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix} ``` Matrix $$C$$ above has $$2$$ rows and $$3$$ columns, so its order is $$2 \times 3$$. Since the number of rows is not equal to the number of columns ($$2 \neq 3$$), matrix $$C$$ is a rectangular matrix. Visible text: Matrix above has rows and columns, so its order is . Since the number of rows is not equal to the number of columns (), matrix is a rectangular matrix. Rectangular matrices can be further distinguished into horizontal matrices and vertical matrices. ### Horizontal Matrix A horizontal matrix is a rectangular matrix with more columns than rows ($$n > m$$). Visible text: A horizontal matrix is a rectangular matrix with more columns than rows (). Example: ```math D = \begin{bmatrix} 1 & 0 & 4 \\ -2 & 3 & 5 \end{bmatrix} ``` Matrix $$D$$ is a horizontal matrix of order $$2 \times 3$$. Visible text: Matrix is a horizontal matrix of order . ### Vertical Matrix A vertical matrix is a rectangular matrix with more rows than columns ($$m > n$$). Visible text: A vertical matrix is a rectangular matrix with more rows than columns (). Example: ```math T = \begin{bmatrix} 7 & 1 \\ 0 & -3 \\ 4 & 2 \end{bmatrix} ``` Matrix $$T$$ is a vertical matrix of order $$3 \times 2$$. Visible text: Matrix is a vertical matrix of order . ## Triangular Matrix A triangular matrix is a square matrix where the elements below or above the main diagonal are zero. ### Upper Triangular Matrix An upper triangular matrix is a square matrix where all elements below the main diagonal are zero. This means $$a_{ij} = 0$$ for every $$i > j$$. Visible text: This means for every . Example: ```math U = \begin{bmatrix} 5 & 2 & -1 \\ 0 & 3 & 7 \\ 0 & 0 & 1 \end{bmatrix} ``` ### Lower Triangular Matrix A lower triangular matrix is a square matrix where all elements above the main diagonal are zero. This means $$a_{ij} = 0$$ for every $$i < j$$. Visible text: This means for every . Example: ```math L = \begin{bmatrix} 2 & 0 & 0 \\ 4 & 6 & 0 \\ -3 & 1 & 9 \end{bmatrix} ``` ## Diagonal Matrix A diagonal matrix is a square matrix where all elements outside the main diagonal are zero. This means $$a_{ij} = 0$$ for every $$i \neq j$$. Elements on the main diagonal can be zero or non-zero. Visible text: This means for every . Elements on the main diagonal can be zero or non-zero. Example: ```math X = \begin{bmatrix} 7 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & 0 \end{bmatrix} ``` Matrix $$X$$ is a diagonal matrix of order $$3 \times 3$$. Visible text: Matrix is a diagonal matrix of order . ## Identity Matrix An identity matrix (denoted by $$I$$ or $$I_n$$) is a diagonal matrix where all elements on the main diagonal are $$1$$. Visible text: An identity matrix (denoted by or ) is a diagonal matrix where all elements on the main diagonal are . Example: Component: MathContainer Children: ```math I_2 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} ``` ```math I_3 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} ``` The identity matrix acts as the neutral element in matrix multiplication. ## Zero Matrix A zero matrix (denoted by $$O$$ or $$O_{m \times n}$$) is a matrix where all elements are zero. Visible text: A zero matrix (denoted by or ) is a matrix where all elements are zero. Example: Component: MathContainer Children: ```math O_{2 \times 2} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} ``` ```math O_{2 \times 3} = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} ``` ## Symmetric Matrix A symmetric matrix is a square matrix that is equal to its transpose ($$A^T = A$$). Visible text: A symmetric matrix is a square matrix that is equal to its transpose (). This means the element $$a_{ij} = a_{ji}$$ for all $$i$$ and $$j$$. Its elements are symmetric with respect to the main diagonal. Visible text: This means the element for all and . Its elements are symmetric with respect to the main diagonal. Example: ```math S = \begin{bmatrix} 1 & 7 & -3 \\ 7 & 2 & 0 \\ -3 & 0 & 5 \end{bmatrix} ``` In matrix $$S$$: Visible text: In matrix : - $$s_{12} = s_{21} = 7$$ - $$s_{13} = s_{31} = -3$$ - $$s_{23} = s_{32} = 0$$ Visible text: - - -