Types of matrices

Matri­ces come in many shapes and pat­terns, and sev­eral of them have spe­cial names because they behave in spe­cial ways. The iden­tity matrix plays the part of the num­ber $1$, the zero matrix the part of $0$, and square matri­ces are the ones that will have deter­mi­nants and inverses later in the course. This les­son intro­duces each type with an exam­ple, and then explains when two matri­ces are equal.

Row and col­umn matri­ces#

A row matrix has exactly one row; its order is $1 \times n$.

$$\begin{bmatrix} 3 & -1 & 0 & 7 \end{bmatrix} \text{ is a } 1 \times 4 \text{ row matrix.}$$

A col­umn matrix has exactly one col­umn; its order is $m \times 1$.

$$\begin{bmatrix} 2 \\ 5 \\ -4 \end{bmatrix} \text{ is a } 3 \times 1 \text{ column matrix.}$$

Square and rec­tan­gu­lar matri­ces#

A square matrix has the same num­ber of rows and columns. A square matrix of order $n \times n$ is often called sim­ply a matrix of order $n$.

A matrix that is not square is called rec­tan­gu­lar.

In a square matrix, the entries $a_{11}, a_{22}, a_{33}, \ldots$ (where the row num­ber equals the col­umn num­ber) form the prin­ci­pal diag­o­nal (or lead­ing diag­o­nal), run­ning from top left to bot­tom right. The sum of the entries on the prin­ci­pal diag­o­nal is called the trace of the matrix.

Exam­ple 1#

Find the trace of $A = \begin{bmatrix} 2 & 5 & 1 \\ 0 & -3 & 4 \\ 7 & 8 & 6 \end{bmatrix}$.

The prin­ci­pal diag­o­nal is $2$, $-3$, $6$. So the trace is $2 + (-3) + 6 = 5$.

Zero matrix#

A matrix in which every entry is $0$ is called a zero matrix or null matrix, writ­ten $O$. It can be of any order:

$$\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}, \qquad \begin{bmatrix} 0 & 0 & 0 \end{bmatrix}$$

Diag­o­nal, scalar and iden­tity matri­ces#

A square matrix in which every entry off the prin­ci­pal diag­o­nal is $0$ is called a diag­o­nal matrix. The diag­o­nal entries them­selves may be any­thing, includ­ing $0$.

$$\begin{bmatrix} 4 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 9 \end{bmatrix} \text{ is a diagonal matrix.}$$

A diag­o­nal matrix whose diag­o­nal entries are all equal is called a scalar matrix.

$$\begin{bmatrix} 5 & 0 \\ 0 & 5 \end{bmatrix} \text{ is a scalar matrix.}$$

A scalar matrix whose diag­o­nal entries are all $1$ is called an iden­tity matrix (or unit matrix), writ­ten $I$, or $I_n$ for order $n$.

$$I_2 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}, \qquad I_3 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$$

Each type is a spe­cial case of the one before: every iden­tity matrix is a scalar matrix, and every scalar matrix is a diag­o­nal matrix.

Tri­an­gu­lar matri­ces#

A square matrix is upper tri­an­gu­lar if every entry below the prin­ci­pal diag­o­nal is $0$, and lower tri­an­gu­lar if every entry above it is $0$.

$$\begin{bmatrix} 1 & 4 & -2 \\ 0 & 3 & 6 \\ 0 & 0 & 5 \end{bmatrix} \text{ is upper triangular;} \qquad \begin{bmatrix} 2 & 0 & 0 \\ 7 & -1 & 0 \\ 3 & 8 & 4 \end{bmatrix} \text{ is lower triangular.}$$

A diag­o­nal matrix is both upper and lower tri­an­gu­lar.

Exam­ple 2#

Name the type of each matrix.

(a) $\begin{bmatrix} 0 & 0 \\ 0 & 0 \\ 0 & 0 \end{bmatrix}$ (b) $\begin{bmatrix} -3 & 0 \\ 0 & -3 \end{bmatrix}$ (c) $\begin{bmatrix} 1 & 0 \\ 6 & 2 \end{bmatrix}$ (d) $\begin{bmatrix} 8 \\ 1 \end{bmatrix}$

(a) A zero matrix of order $3 \times 2$ (also rec­tan­gu­lar).

(b) A scalar matrix (also diag­o­nal, square, and both upper and lower tri­an­gu­lar).

(c) A lower tri­an­gu­lar matrix: the only entry above the diag­o­nal is $0$.

(d) A col­umn matrix of order $2 \times 1$.

Equal­ity of matri­ces#

Two matri­ces $A$ and $B$ are equal, writ­ten $A = B$, when

1. they have the same order, and
2. every entry of $A$ equals the entry in the same place in $B$: $a_{ij} = b_{ij}$ for all $i$ and $j$.

So $\begin{bmatrix} 1 & 2 \end{bmatrix}$ and $\begin{bmatrix} 1 \\ 2 \end{bmatrix}$ are not equal: they hold the same num­bers but have dif­fer­ent orders.

One matrix equa­tion is really a set of ordi­nary equa­tions, one for each posi­tion. This is very use­ful for find­ing unknowns.

Exam­ple 3#

Find $x$ and $y$ if $\begin{bmatrix} x + y & 2 \\ 5 & x - y \end{bmatrix} = \begin{bmatrix} 6 & 2 \\ 5 & 2 \end{bmatrix}$.

Equate the entries in the same places: $x + y = 6$ and $x - y = 2$. Adding, $2x = 8$, so $x = 4$; then $y = 2$.

Exam­ple 4#

Find $a$, $b$, $c$, $d$ if $\begin{bmatrix} 2a + b & a - 2b \\ 5c - d & 4c + 3d \end{bmatrix} = \begin{bmatrix} 4 & -3 \\ 11 & 24 \end{bmatrix}$.

From the first row: $2a + b = 4$ and $a - 2b = -3$. Mul­ti­ply the first by $2$ and add: $5a = 5$, so $a = 1$, and then $b = 4 - 2 = 2$.

From the sec­ond row: $5c - d = 11$, so $d = 5c - 11$. Sub­sti­tute into $4c + 3d = 24$:

$$4c + 15c - 33 = 24 \implies 19c = 57 \implies c = 3$$

Then $d = 15 - 11 = 4$. So $a = 1$, $b = 2$, $c = 3$, $d = 4$.

Prac­tice#

1. Write a row matrix of order $1 \times 3$ and a col­umn matrix of order $2 \times 1$.
2. Find the trace of $\begin{bmatrix} 7 & 2 \\ -4 & 9 \end{bmatrix}$.
3. Find the trace of $\begin{bmatrix} 1 & 0 & 3 \\ 5 & -6 & 2 \\ 4 & 8 & 10 \end{bmatrix}$.
4. Name the type of $\begin{bmatrix} 3 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3 \end{bmatrix}$ as pre­cisely as you can.
5. Name the type of $\begin{bmatrix} 2 & 5 \\ 0 & -1 \end{bmatrix}$.
6. Is every diag­o­nal matrix a scalar matrix? Give a rea­son.
7. Write the iden­tity matrix of order $4$.
8. Are $\begin{bmatrix} 3 & 0 \end{bmatrix}$ and $\begin{bmatrix} 3 & 0 & 0 \end{bmatrix}$ equal? Give a rea­son.
9. Find $x$ and $y$ if $\begin{bmatrix} 2x & 3 \\ 1 & y + 4 \end{bmatrix} = \begin{bmatrix} 10 & 3 \\ 1 & 1 \end{bmatrix}$.
10. Find $p$ and $q$ if $\begin{bmatrix} p + q \\ p - q \end{bmatrix} = \begin{bmatrix} 9 \\ 1 \end{bmatrix}$.
11. Find $a$, $b$, $c$ if $\begin{bmatrix} a & b + 1 \\ 2c & 6 \end{bmatrix}$ is a scalar matrix.

Answers#