Types of Matrices

4.4.1 Types of Matrices

1. Row matrix. Only one row; order $1\times n$. E.g. $[-1\ \ 2]_{1\times2}$, $[0\ \ -3\ \ 5]_{1\times3}$.
2. Column matrix. Only one column; order $m\times1$. E.g. $\begin{bmatrix}1\\0\\2\end{bmatrix}_{3\times1}$. Note: a single-element matrix like $[5]_{1\times1}$ is simultaneously a row and a column matrix.
3. Zero (null) matrix. Every element is $0$, denoted $O$. E.g. $O=\begin{bmatrix}0&0&0\\0&0&0\\0&0&0\end{bmatrix}_{3\times3}$.
4. Square matrix. Equal number of rows and columns (order $n\times n$, "square matrix of order $n$"). For a square matrix $A=[a_{ij}]_{n\times n}$: the entries $a_{11},a_{22},\dots,a_{nn}$ are the diagonal elements; entries with $i\ne j$ are non-diagonal; entries with $i<j$ lie above the diagonal, entries with $i>j$ lie below it. (Diagonal elements are only defined for a square matrix.)
5. Diagonal matrix. A square matrix in which every non-diagonal element is $0$. E.g. $\begin{bmatrix}5&0&0\\0&9&0\\0&0&3\end{bmatrix}$.
6. Scalar matrix. A diagonal matrix whose diagonal elements are all equal. E.g. $\begin{bmatrix}5&0&0\\0&5&0\\0&0&5\end{bmatrix}$.
7. Unit (identity) matrix. A scalar matrix whose diagonal elements are all $1$; denoted $I_n$. E.g. $I_3=\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}$, $I_2=\begin{bmatrix}1&0\\0&1\end{bmatrix}$. Note: every identity matrix is a scalar matrix, but not every scalar matrix is an identity matrix — a scalar matrix is a scalar multiple of the identity matrix. Every scalar matrix is a diagonal matrix, but not every diagonal matrix is a scalar matrix.
8. Upper triangular matrix. Every entry below the diagonal is $0$ ($a_{ij}=0$ for $i>j$). E.g. $\begin{bmatrix}9&3&4\\0&0&1\\0&0&-2\end{bmatrix}$ — wait, more simply: every entry strictly below the leading diagonal is zero.
9. Lower triangular matrix. Every entry above the diagonal is $0$ ($a_{ij}=0$ for $i<j$).
10. Triangular matrix. Upper triangular OR lower triangular. Note: diagonal, scalar, unit and null (square) matrices are automatically triangular too.
11. Symmetric matrix. A square matrix $A=[a_{ij}]$ with $a_{ij}=a_{ji}$ for all $i,j$ — i.e. it looks the same reflected across the main diagonal. E.g. $\begin{bmatrix}a&h&g\\h&b&f\\g&f&c\end{bmatrix}$. Note: every scalar matrix is symmetric; a null square matrix is symmetric.
12. Skew-symmetric matrix. A square matrix with $a_{ij}=-a_{ji}$ for all $i,j$. Setting $i=j$ forces $a_{ii}=-a_{ii}\Rightarrow2a_{ii}=0\Rightarrow a_{ii}=0$ — so every diagonal element of a skew-symmetric matrix is $0$. E.g. $\begin{bmatrix}0&5\\-5&0\end{bmatrix}$. Note: a null square matrix is also (trivially) skew-symmetric.
13. Determinant of a matrix. Defined only for a square matrix: replace the square brackets of $A$ by vertical bars to get $|A|$ or $\det(A)$.
14. Singular / non-singular matrix. A square matrix $A$ is singular if $|A|=0$; otherwise it is non-singular.
15. Transpose of a matrix. The matrix obtained by interchanging the rows and columns of $A$, denoted $A^T$ (or $A'$) — if $A$ is $m\times n$ then $A^T$ is $n\times m$, and $(A^T)_{ij}=A_{ji}$. Remarks (used constantly later): if $A$ is symmetric, $A=A^T$; if $A$ is skew-symmetric, $A^T=-A$.

Worked Examples

Activity. Construct $A=[a_{ij}]_{2\times2}$ where $a_{ij}=\left(\dfrac{i+j}{2}\right)^2$: substituting $(i,j)=(1,1),(1,2),(2,1),(2,2)$ gives $a_{11}=\left(\dfrac{2}{2}\right)^2=1$, $a_{12}=\left(\dfrac32\right)^2=\dfrac94$, $a_{21}=\left(\dfrac32\right)^2=\dfrac94$, $a_{22}=\left(\dfrac42\right)^2=4$, so $A=\begin{bmatrix}1&\tfrac94\\\tfrac94&4\end{bmatrix}$ — and since $a_{12}=a_{21}$ this $A$ is symmetric.

Example 1. Show $\begin{vmatrix}x+y&y+z&z+x\\z&x&y\\1&1&1\end{vmatrix}$-style matrix (built from $x,y,z$-sums) is singular.

Step 1: $|A|=(x+y)(y-x)-(y+z)(y-z)+(z+x)(x-z)$ style expansion $=y^2-x^2-y^2+z^2+x^2-z^2=0$. …