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Matrix

Matrix
A rectangular arrangement of numbers arranged in rows and columns enclosed with large brackets or round brackets is known as matrix. Usually, matrix is represented by a capital letter. It’s plural is matrices. E.g.
A = $\begin{bmatrix}a&b\\c&d\\\end{bmatrix}$
B = $\left(\begin{matrix} 1&2&3\\1&4&9\\\end{matrix}\right)$

Types of matrix

1. Row matrix
A matrix having only one row is called a row matrix. For examples A = [2 4 6], B = [3, 4] are row matrices.
2. Column matrix
A matrix having only one column is called a column matrix. For examples A = $\begin{bmatrix} 5\\6 \\\end{bmatrix}$ and B = $\begin{bmatrix}p\\q\\r\\\end{bmatrix}$
3. Zero or null matrix
A matrix with all element zero is called a zero or null matrix. For examples $\begin{bmatrix}0&0\\0&0\\\end{bmatrix}$ and $\begin{bmatrix}0&0&0\\0&0&0\\0&0&0\\\end{bmatrix}$ are null matrices of order 2 $\times$ 2 and 3 $\times$ 3 respectively.
4. Square matrix
A matrix in which the number of rows and columns are equal is called a square matrix. For e.g. A = $\begin{bmatrix}2&4\\4&16\\\end{bmatrix}$ and B = $\begin{bmatrix}1&5&8\\3&4&6\\9&4&2\\\end{bmatrix}$ are square matrices.
5. Diagonal matrix
A square matrix in which elements in the leading diagonal are non-zero and zero is called a diagonal matrix. For example C = $\begin{bmatrix} 3&0\\0&5\\\end{bmatrix}$ and $\begin{bmatrix} 4&0&0\\0&8&0\\0&0&16\\\end{bmatrix}$
6. Scalar matrix
A digital matrix in which all the diagonal elements are equal, is called a scalar matrix. For examples A = $\begin{bmatrix}4&0\\0&4\\\end{bmatrix}$ and B = $\begin{bmatrix}x&0&0\\0&x&0\\0&0&x\\\end{bmatrix}$ are scalar matrices.
7. Identity or unit matrix
A square matrix in which the elements in the leading diagonal are 1 and rest are zero, is called a unit matrix or an identity matrix. It is denoted by I. For examples I = $\begin{bmatrix} 1&0\\0&1\\\end{bmatrix}$ and I = $\begin{bmatrix} 1&0&0\\0&1&0\\ 0&0&1\\\end{bmatrix}$ are unit matrices of order 2 $\times$ 2 and 3 $\times$ 3 respectively.
8. Equal matrix
Two matrices are said to be equal, if they are of the same order and their corresponding elements are equal. For examples if A = $\begin{bmatrix}2&4\\6&8\\\end{bmatrix}$ and B = $\begin{bmatrix}2&4\\6&8\\\end{bmatrix}$ then, A = B.
Thus, $\begin{bmatrix} p&q\\r&s\\\end{bmatrix}$ = $\begin{bmatrix}6&8\\3&6\\\end{bmatrix}$ if and only if, p = 6, q = 8, r = 3 and s = 6.

Examples

• Construct a 2 $\times$ 2 matrix whose elements are given by aij = $\frac{i}{2j}$
Solution:
Let a 2 $\times$ 2 matrix is $\begin{bmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\\\end{bmatrix}$
Now,
a11 = $\frac{1}{2\times1}$ = $\frac{1}{2}$
a12 = $\frac{1}{2\times2}$ = $\frac{1}{4}$
a21 = $\frac{2}{2\times1}$ = $\frac{2}{2}$ = 1
a22 = $\frac{2}{2\times2}$ = $\frac{2}{4}$ = $\frac{1}{2}$
$\therefore$ The required matrix is $\begin{bmatrix} \frac{1}{2}& \frac{1}{4}\\1&\frac{1}{2}\\\end{bmatrix}$
• Find the value of x, y, if $\left(\begin{matrix} x + y \\ x – y \\\end{matrix}\right)$ = $\left(\begin{matrix} 5\\3\\\end{matrix}\right)$.
Solution:
x + y = 5 -------(i)
x – y = 3 -------(ii)
$\begin{matrix} &x& +& y& =& 5&\\ &x& -& y& =& 3&\\ (+)&&&&&&\\ \hline &2x&&&=&8\\ \end{matrix}$
or, x = $\frac{8}{2}$
$\therefore$ x = 4
Putting the value of x = 4 in equation (i)
x + y = 5
or, 4 + y = 5
or, y = 5 – 4
$\therefore$ y = 1
$\therefore$ x = 4 and y = 1

Transpose of a matrix
Let A = $\begin{bmatrix}2&3&5\\6&7&0\\\end{bmatrix}$ be a 2 $\times$ 3 matrix. Interchanging the rows and columns of A we get a new matrix of order 3 $\times$ 2. The new matrix is $\begin{bmatrix}2&6\\3&7\\5&0\\\end{bmatrix}$ and is denoted by A’ or AT.
$\therefore$ AT = $\begin{bmatrix}2&6\\3&7\\5&0\\\end{bmatrix}$. Thus, for a given matrix A, a new matrix formed by interchanging its rows and columns is called the transpose of A.

Properties of Transpose

• (AT)T = A
• (A + B)T = AT + BT
• (KA)T = KAT

Examples

• If A = $\begin{bmatrix}1&3\\2&4\\\end{bmatrix}$, B = $\begin{bmatrix}0&3\\2&1\\\end{bmatrix}$ and C = $\begin{bmatrix}5&7\\6&8\\\end{bmatrix}$. Find A – B + C.
Solution:
A – B + C = $\begin{bmatrix}1&3\\2&4\\\end{bmatrix}$ - $\begin{bmatrix}0&3\\2&1\\\end{bmatrix}$ + $\begin{bmatrix}5&7\\6&8\\\end{bmatrix}$
= $\begin{bmatrix}1-0+5&3-3+7\\2-2+6&4-1+8\\\end{bmatrix}$
= $\begin{bmatrix}6&7\\6&11\\\end{bmatrix}$
• If A = $\begin{bmatrix}1&2\\3&4\\\end{bmatrix}$, B = $\begin{bmatrix}1&0\\2&3\\ \end{bmatrix}$, then prove the following A + B = B + A
Solution:
L.H.S. = A + B = $\begin{bmatrix}1&2\\3&4\\\end{bmatrix}$ + $\begin{bmatrix}1&0\\2&3\\\end{bmatrix}$
= $\begin{bmatrix}1+1&2+0\\3+2&4+3\\\end{bmatrix}$
= $\begin{bmatrix}2&2\\5&7\\\end{bmatrix}$
R.H.S. = B + A = $\begin{bmatrix}1&0\\2&3\\\end{bmatrix}$ + $\begin{bmatrix}1&2\\3&4\\\end{bmatrix}$
= $\begin{bmatrix}1+1&0+2\\2+3&3+4\\\end{bmatrix}$
= $\begin{bmatrix}2&2\\5&7\\\end{bmatrix}$
$\therefore$ L.H.S. = R.H.S.

Multiplication of Matrix

Examples

• If A = $\left(\begin{matrix}1&-1\\-3&3\\\end{matrix}\right)$ and B = $\left(\begin{matrix}2&-5\\2&-5\\\end{matrix}\right)$. Show that AB is a null matrix.
Solution:
AB = $\left(\begin{matrix}1&-1\\-3&3\\\end{matrix}\right)$$\left(\begin{matrix}2&-5\\2&-5\\\end{matrix}\right)$
= $\left(\begin{matrix}1\times 2 + (-1)\times2& 1\times(-5) + (-1)\times(-5)\\ -3\times2 + 3\times2&-3\times(-5) + 3\times(-5)\\\end{matrix}\right)$
= $\left(\begin{matrix}2-2&-5+5\\-6+6&15-15\\\end{matrix}\right)$
= $\left(\begin{matrix}0&0\\0&0\\\end{matrix}\right)$
$\therefore$ AB is a null matrix.
• If A = $\left(\begin{matrix}3&4\\2&3\\\end{matrix}\right)$ and B = $\left(\begin{matrix} 3&-4\\-2&3\\\end{matrix}\right)$ then, Prove that AB is an identity matrix.
Solution:
AB = $\left(\begin{matrix}3&4\\2&3\\\end{matrix}\right)$$\left(\begin{matrix}3&-4\\-2&3\\\end{matrix}\right)$
= $\left(\begin{matrix}3\times3 + 4\times(-2)&3\times(-4) + 4\times3\\2\times3 + 3\times(-2)&2\times(-4) + 3\times3\\\end{matrix}\right)$
= $\left(\begin{matrix}9-8&-12+12\\6-6&-8+9\\\end{matrix}\right)$
= $\left(\begin{matrix}1&0\\0&1\\\end{matrix}\right)$
$\therefore$ The matrix AB is a identity matrix.