# Matrices with Examples and Questions with Solutions

Examples and questions on matrices along with their solutions are presented .

## Definition of a Matrix

The following are examples of matrices (plural of matrix). An m � n (read 'm by n') matrix is an arrangement of numbers (or algebraic expressions ) in m rows and n columns . Each number in a given matrix is called an element or entry .
A
zero matrix has all its elements equal to zero.

Example 1
The following matrix has 3 rows and 6 columns. The order (or dimensions or size) of a matrix indicates the number of rows and the number of columns of the matrix. In this example, the order of the matrix is 3 � 6 (read '3 by 6').

## Matrix entry (or element)

   

The entry (or element) in a row i and column j of a matrix A (capital letter A) is denoted by the symbol $$(A)_{ij}$$ or $$a_{ij}$$ (small letter a).

Example 2

In the matrix A shown below, $$a_{11} = 5$$, $$a_{12} = 2$$, etc ... or $$(A)_{11} = 5$$, $$(A)_{12} = 2$$, etc ... $\textbf{A} = \begin{bmatrix} 5 & 2 & 7 & -3 \\ -9 & -2 & -7 & 11\\ \end{bmatrix} = \begin{bmatrix} a_{11} & a_{12} & a_{13} & a_{14} \\ a_{21} & a_{22} & a_{23} & a_{24} \\ \end{bmatrix}$

## Square Matrix

A square matrix has the number of rows equal to the number of columns.

Example 3

For each matrix below, determine the order and state whether it is a square matrix.
$a) \begin{bmatrix} -1 & 1 & 0 & 3 \\ 4 & -3 & -7 & -9\\ \end{bmatrix} \;\;\;\; b) \begin{bmatrix} -6 & 2 & 0 \\ 3 & -3 & 4 \\ -5 & -11 & 9 \end{bmatrix} \;\;\;\; \\ c) \begin{bmatrix} 1 & -2 & 5 & -2 \end{bmatrix} \;\;\;\; d) \begin{bmatrix} -2 & 0 \\ 0 & -3 \end{bmatrix} \;\;\;\; e) \begin{bmatrix} 3 \end{bmatrix}$
Solutions
a) order: 2 � 4. Number of rows and columns are not equal therefore not a square matrix.
b) order: 3 � 3. Number of rows and columns are equal therefore this matrix is a square matrix.
c) order: 1 � 4. Number of rows and columns are not equal therefore not a square matrix. A matrix with one row is called a row matrix (or a row vector).
d) order: 2 � 2. Number of rows and columns are equal therefore this is square matrix.
e) order: 1 � 1. Number of rows and columns are equal therefore this matrix is a square matrix.

## Identity Matrix

An identity matrix In is an n�n square matrix with all its element in the diagonal equal to 1 and all other elements equal to zero.
Example 4
The following are all identity matrices. $I_1= \begin{bmatrix} 1 \\ \end{bmatrix} \quad , \quad I_2= \begin{bmatrix} 1 & 0\\ 0& 1 \end{bmatrix} \quad , \quad I_3= \begin{bmatrix} 1 & 0 & 0\\ 0& 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$

## Diagonal Matrix

A diagonal matrix is a square matrix with all its elements (entries) equal to zero except the elements in the main diagonal from top left to bottom right. $A = \begin{bmatrix} 6 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & 2 \end{bmatrix}$

## Triangular Matrix

An upper triangular matrix is a square matrix with all its elements below the main diagonal equal to zero. Matrix U shown below is an example of an upper triangular matrix. A lower triangular matrix is a square matrix with all its elements above the main diagonal equal to zero. Matrix L shown below is an example of a lower triangular matrix.
$$U = \begin{bmatrix} 6 & 2 & -5 \\ 0 & -2 & 7 \\ 0 & 0 & 2 \end{bmatrix} \qquad L = \begin{bmatrix} 6 & 0 & 0 \\ -2 & -2 & 0 \\ 10 & 9 & 2 \end{bmatrix}$$

## Transpose of a Matrix

The transpose of an m�n matrix $$A$$ is denoted $$A^T$$ with order n�m and defined by $(A^T)_{ij} = (A)_{ji}$ Matrix $$A^T$$ is obtained by transposing (exchanging) the rows and columns of matrix $$A$$.
Example 5
$\begin{bmatrix} 6 & 0 \\ -2 & -2\\ 10 & 9 \end{bmatrix} ^T = \begin{bmatrix} 6 & -2 & 10 \\ 0 & -2 &9\\ \end{bmatrix}$ Transpose a matrix an even number of times and you get the original matrix: $$((A)^T)^T = A$$. Transpose matrix an odd number of times and you get the transpose matrix: $$(((A)^T)^T)^T = A^T$$.
The transpose of any square diagonal matrix is the matrix itself. $\begin{bmatrix} 3 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & 6 \end{bmatrix} ^T = \begin{bmatrix} 3 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & 6 \end{bmatrix}$

## Symmetric Matrix

A square matrix is symmetric if its elements are such that $$A_{ij} = A_{ji}$$ in other words $$A$$ is symmetric if $$A = A^T$$.
Example 6
Symmetric matrices $\begin{bmatrix} 4 & -2 & 1 \\ -2 & 5 & 7 \\ 1 & 7 & 8 \end{bmatrix}$

## Questions on Matrices: Part A

Given the matrices: $A = \begin{bmatrix} -1 & 23 & 10 \\ 0 & -2 & -11 \\ \end{bmatrix} ,\quad B = \begin{bmatrix} -6 & 2 & 10 \\ 3 & -3 & 4 \\ -5 & -11 & 9 \\ 1 & -1 & 9 \end{bmatrix} ,\quad C = \begin{bmatrix} -3 & 2 & 9 & -5 & 7 \end{bmatrix} \\ D = \begin{bmatrix} -2 & 6 \\ -5 & 2\\ \end{bmatrix} ,\quad E = \begin{bmatrix} 3 \end{bmatrix} ,\quad F = \begin{bmatrix} 3 \\ 5 \\ -11 \\ 7 \end{bmatrix} ,\quad G = \begin{bmatrix} -6 & -4 & 23 \\ -4 & -3 & 4 \\ 23 & 4 & 9 \\ \end{bmatrix}$
a) What is the dimension of each matrix?
b) Which matrices are square?
c) Which matrices are symmetric?
d) Which matrix has the entry at row 3 and column 2 equal to -11?
e) Which matrices has the entry at row 1 and column 3 equal to 10?
f) Which are column matrices?
g) Which are row matrices?
h) Find $$A^T , C^T , E^T , G^T$$.

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## Questions on Matrices: Part B

1) Given the matrices: $A = \begin{bmatrix} 23 & 10 \\ 0 & -11 \\ \end{bmatrix} ,\quad B = \begin{bmatrix} -6 & 0 & 0 \\ -1 & -3 & 0 \\ -5 & 3 & -9 \\ \end{bmatrix} ,\quad C = \begin{bmatrix} -3 & 0\\ 0 & 2 \end{bmatrix} \\ ,\quad D = \begin{bmatrix} -7 & 3 & 2 \\ 0 & 2 & 4 \\ 0 & 0 & 9 \\ \end{bmatrix} ,\quad E = \begin{bmatrix} 12 & 0 & 0 \\ 0 & 23 & 0 \\ 0 & 0 & -19\\ \end{bmatrix}$
a) Which of the above matrices are diagonal?
b) Which of the above matrices are lower triangular?
c) Which of the above matrices are upper triangular?

## Solutions to the Questions in Part A

a) A: 2 � 3, B: 4 � 3, C: 1 � 5, D: 2 � 2, E: 1 � 1, F: 4 � 1, G: 3 � 3,
b) D, E and G
c) E and G
d) B
e) A and B
f) E and F
g) E and C
h) $A^T = \begin{bmatrix} -1 & 0 \\ 23 & -2 \\ 10 & -11 \end{bmatrix} ,\quad C^T = \begin{bmatrix} -3 \\ 2\\ 9\\-5\\7 \end{bmatrix} ,\quad E^T = \begin{bmatrix} 3 \end{bmatrix} ,\quad G^T = \begin{bmatrix} -6 & -4 & 23\\ -4 & -3 & 4\\ 23 & 4 & 9 \end{bmatrix}$

a) C and E
b) B
c) A and D