Matrices

 Matrices are arrangements of numbers in rectangular arrays. The plural for Matrix is Matrices.

$\begin{bmatrix}
a_{11} & a_{12} & a_{13} & . &.  & . &a_{1n} \\
a_{21} & a_{22} & a_{23} & . &.  & . &a_{2n} \\
 .& . & .&  .&. & . &.\\
 .& . &  .& . & . & . &. \\
. & . & . & . & . & . & .\\
a_{m1} &a_{m2}  & a_{m3} & . &.  & . & a_{mn}
 
\end{bmatrix}$

The entries are done in m rows (horizontal lines) and n columns (vertical lines). Each entry $a_{ij}$ is generally a number and is also called an element of the matrix.A matrix with m rows and n columns is called a m x n (read as "m by n") Matrix.  The size of a matrix is indicated by (m x n). If each entry in a matrix is a real number, then the matrix is called a real matrix.
 

Notations for a Matrix

  1. A matrix can be denoted by uppercase letters like A,B,C.....
  2. A matrix can be denoted by writing the representative element inside a bracket like $[a_{ij}]$, $[b_{ij}]$.......
  3. A matrix can be written in a rectangular array.

Matrix Equality

Two Matrices of equal sizes are equal if each entry in one matrix is equal to the corresponding entry in the other matrix.
Matrices $A=[a_{ij}]$ and $B=[b_{ij}]$  of size (m x n) are equal if
$[a_{ij}]=[b_{ij}]$ for 1 ≤ i ≤ m and 1 ≤ j ≤ n.

Example:
Consider the matrices

$A=\begin{bmatrix}
2 &-1 \\
 0& 3
\end{bmatrix}$
  $B=\begin{bmatrix}
2 &-1 \\
 0& 3
\end{bmatrix}$
 Matrices A and B are equal as they are of equal size (2 x 2)
and each element in Matrix A is equal to corresponding element in B
 $C=\begin{bmatrix}
1 & 2 & 3
\end{bmatrix}$
 $D=\begin{bmatrix}
1\\
2\\
3
\end{bmatrix}$
 Even though the same elements are used in forming the matrices
C and D, they are of different sizes (1 x 3) and (3 x 1)


Row and Column Matrices

A matrix with only one row as the Matrix C in the above example is called a row matrix. A general row matrix is of size (1 x n).
Similarly a matrix with only one column as Matrix D is known as a column matrix. The size of a general column matrix is (m x 1).
Any matrix can be partitioned into row and column matrices.

Squrare Matrices

Square Matrix

If a matrix has equal number of rows and columns then it is called a square matrix. A general square matrix is of size (n x n).

Example:
$\begin{bmatrix}
1 & -2\\
3 & 0
\end{bmatrix}$

Diagonal matrix


The entries in a diagonal matrix is all zero except the diagonal elements.

Example:
$\begin{bmatrix}
2 & 0 &0 \\
0 & -1 &0 \\
0 & 0 & 1
\end{bmatrix}$
If all the elements of a diagonal matrix are equal, then it is called a scalar matrix.

Unit or identity matrix


Unit matrix is a scalar (diagonal matrix) with all diagonal elements as unity.

Example:
$\begin{bmatrix}
1 & 0 &0 \\
0 & 1 &0 \\
0 & 0 & 1
\end{bmatrix}$    
 $\begin{bmatrix}
1 & 0 & 0 &0\\
 0& 1 &0  & 0\\
 0& 0 & 1 &0\\
 0&  0& 0 & 1
\end{bmatrix}$


Matrix Operations


Matrix arithmetic consists of addition, subtraction, Scalar multiplication and Matrix multiplication in a manner similar to operations on numbers.

Matrix Addition


Matrix is addition is defined only for matrices of same size as follows:
If A = $[a_{ij}]$ and B = $[b_{ij}]$ are two matrices of size (m x n) then
A + B = $[a_{ij}+b_{ij}]$

Addition of two matrices of different sizes is not defined.

Example:

$\begin{bmatrix}
3 & 0 &-1 \\
-2 & 5 & 3
\end{bmatrix}$
+
$\begin{bmatrix}
-1 & 2 &3 \\
0 & -2 & 1
\end{bmatrix}$ 
 = $ \begin{bmatrix}
2 & 2 &2 \\
-2 & 3 & 4
\end{bmatrix}$

Scalar Multiplication


A scalar refers to a real number.

If A = $[a_{ij}]$ is a m x n  matrix and c a scalar(a real number), then the scalar multiple of A by c is a m x n matrix given by
cA = $[ca_{ij}]$.

For example if A is the column matrix $\begin{bmatrix}
-2\\
1\\
\end{bmatrix}$             and c = 3, then cA is a 3 x 1 matrix
cA = 3A = $\begin{bmatrix}
-6\\
3\\
9
\end{bmatrix}$.

Matrix subtraction


Matrix subtraction is defined combining the definitions of matrix addition and scalar multiplication.
If A = $[a_{ij}]$  then -A represent the scalar product (-1)A. Hence -A = $[-a_{ij}]$.

If A and B are matrices of same size then the difference A - B is the sum of A and (-1)B.
Thus A - B = A + (-1)B.

Example :

If A = $\begin{bmatrix}
1 & 2\\
2 & 1
\end{bmatrix}$  and B = $\begin{bmatrix}
-3 & -2\\
4 & 2
\end{bmatrix}$, Find 2A - B.
2A-B
=
 2$\begin{bmatrix}
1 & 2\\
2 & 1
\end{bmatrix}$
 +
 (-1)$\begin{bmatrix}
-3 & -2\\
4 & 2
\end{bmatrix}$
  =  $\begin{bmatrix}
2 &4 \\
4 & 2
\end{bmatrix}$
 +  $\begin{bmatrix}
3 &2 \\
-4 & -2
\end{bmatrix}$
   =  $\begin{bmatrix}
5 &6 \\
0 & 0
\end{bmatrix}$
   

Properties of Matrix addition and scalar multiplication


If A,B and C are m x n Matrices and c and d are scalars, then the following properties hold good.

1. A + B = B + A Commutative property of addition
2. A + (B + C) = (A + B) + C
Associative property of addition
3. (cd)A =c(dA) Associative property of scalar multiplication
4. 1A = 1 Multiplicative identity
5. c(A + B) = cA + cB Distributive property
6.(c + d)A = cA + dA Distributive property

Matrix Multiplication

If matrices are useful to display information, matrix multiplication is used to manipulate and organize the information for the requirement.

                                                              Definition - Matrix multiplication

If A = $[a_{ij}]$ is a m x n matrix and B = $[b_{ij}]$ is a n x p matrix, then product AB = $[c_{ij}]$ is the m x p matrix where
$c_{ij}=\sum_{k=1}^{n}a_{ik}b_{kj}=a_{i1}b_{1k}+a_{i2}b_{2k}+a_{i3}b_{3k}+.......+a_{in}b_{nj}$

The element in the $i^{th}$ row and the $j^{th}$ column of the product AB is got by multiplying the entries in the $i^{th}$
row of matrix A by the corresponding entries in the $j^{th}^ column of Matrix B and adding up the results.
 
Note: You cannot multiply any two matrices or even any matrices of same size. The condition that makes the matrix multiplication AB possible is
Number of columns in the first matrix A = Number of rows in the second matrix B.

Example:
Find the product AB if it exist for A = $\begin{bmatrix}
-1 & 3\\
4& -5\\
0& 2
\end{bmatrix}$  and B = $\begin{bmatrix}
1 & 2\\
 0& 7
\end{bmatrix}$

A is a 3 x 2 matrix and B a 2 x 2 matrices.
Number of columns in matrix A = number of columns in matrix B = 2
Hence A and B are compatible for multiplication and the product AB is a matrix of size 3 x 2.
If AB = $\begin{bmatrix}
c_{11} & c_{12}\\
c_{21}& c_{22}\\
c_{31}& c_{32}
\end{bmatrix}$ then
$c_{11}= (-1)1 +3(0) =-1$              First row in A and First column in B
$c_{12}= (-1)2 +3(7) =19$               First row in A and second column in B
$c_{21}= 4(1) -5(0)=4$                    Second row in A and first column in B
$c_{22}= 4(2)-5(7)=-27$               Second row in A and second column in B
$c_{31}= 0(1)+2(0)=0$                    Third row in A and first column in B
$c_{32}= 0(2)+2(7)=14$                  Third row in A and second column in B

AB = $\begin{bmatrix}
-1 & 19\\
4 & -27\\
0& 14
\end{bmatrix}$

Properties of Matrix multiplication


If A, B and C are matrices compatible for the products defined and c is a scalar, then the following properties hold good.

1. A(BC) = (AB)C
Associative property
2. A(B + C) = AB + AC
Left distributive property
3. (A + B)C = AC + BC
Right distributive property
4. c(AB) = (cA)B = A(cB)
Associative property




Transpose of a matrix

The transpose of a matrix is formed by writing its columns as rows and rows as columns. It is denoted by $A^{T}$
If A = $[a_{ij}]$ is a m x n matrix then its transpose $A^{T}$ = $[a_{ji}]$ is a n x m matrix.

Example:
$A =\begin{bmatrix}
0 & 2 & 1\\
1 & 4 & -1
\end{bmatrix}$   A is a 2 x 3 matrix. Then its transpose $A^{T}$ is a 3 x 2 matrix.
$A^{T}= \begin{bmatrix}
0 &1 \\
2 &4 \\
 1& -1
\end{bmatrix}$

Symmetric Matrix


A matrix is symmetric if A = $A^{T}$. This means a symmetric matrix is a square matrix.
If A = $[a_{ij}]$ is a symmetric matrix, then $a_{ij}=a_{ji}$ for all i ≠ j.

Example:
$\begin{bmatrix}
1 & 2 & -2\\
2 & 1 &0 \\
-2 & 0 & 3
\end{bmatrix}$

Skew Symmetric Matrix


A matrix B is called skew symmetric if B = $-B^{T}$
If A = $[a_{ij}]$ is a skew symmetric matrix, then $a_{ij}=-a_{ji}$ for all i ≠ j. and $a_{ij}=0$ when i = j.
This means the diagonal

Example:
$\begin{bmatrix}
0 & -1 &-2 \\
1 & 0 & -1\\
 2&1 & 0
\end{bmatrix}$

Properties of Transposes


$(A^{T})^{T}=A$
Transpose of the transpose matrix is the matrix.
$(A+B)^{T}=A^{T}+B^{T}$ Transpose of a sum is the sum of the transposes.
${(cA)}^{T}=c(A)^{T}$ Scalar Multiple of a transpose
$(AB)^{T}=(B)^{T}(A)^{T}$ Transpose of a product.


Deterrminant of a Matrix

Determinant is a function that associates a square matrix to a real number.
Determinant of a square matrix A is denoted by |A|.

Determinant of a 2 x 2 matrix.
The determinant of a matrix $A=\begin{vmatrix}
a_{11} & a_{12}\\
a_{21} & a_{22}
\end{vmatrix}$ is given by

det(A) = |A| = $a_{11}a_{22}-a_{21}a_{12}$

Example:
$A=\begin{vmatrix}
 2&-3 \\
1 & 2
\end{vmatrix}$

|A| = 2(2) -1(-3) = 4 + 3 =7

Minor of a Matrix


Let A = $[a_{ij}]$ be a square Matrix. Then the minor $M_{ij}$ of the element $a_{ij}$ is the determinant of a Matrix obtained by deleting the $i^{th}$ row and the $j^{th}$ column of A.
Example:
$A=\begin{bmatrix}
0 &2  & 1\\
3 &  -1&2 \\
 4& 0 & 1
\end{bmatrix}$Find the minor of the element 3
A is  a 3 x 3 Matrix. The element 3 is row 2 and column 1. Removing row 2 and column from |A|, the minor of 3 is
Minor of 3 = $M_{21}$=$\begin{vmatrix}
2 & 1\\
0 & 1
\end{vmatrix}$.=2(1)-0(1) =2

Cofactor of a Matrix


The cofactor $C_{ij}$ of the element $a_{ij}$  is
$C_{ij}=(-1)^{i+j}M_{ij}$
For the example given above the element 3 is row 2 and column 1. i =2 and j =1. Hence
Cofactor of 3 = $C_{21}=(-1)^{2+1}M_{21}=(-1)^{3}(2)=-2$

Expansion of Determinant using cofactors


Using Cofactors the determinant of a Square Matrix of order n can be defined. The determinant can be evaluated along any row or any column of the determinant.

Expansion of |A| along the $i^{th}$ row
$det(A)=|A|=\sum_{j=1}^{n}a_{ij}C_{ij}= a_{i1}C_{i1}+a_{i2}C_{i2}+.......a_{in}C_{in}$
Expansion of |A| along the $j^{th}$ column
$det(A)=|A|=\sum_{i=1}^{n}a_{ij}C_{ij}= a_{1j}C_{1j}+a_{2j}C_{2j}+.......a_{nj}C_{nj}$



Ajoint of a Matrix

If C is the matrix formed by the cofactors of Matrix A, then the adjoint of the Matrix A is the transpose of Matrix C.

Let A = $\begin{bmatrix}
a_{11} & a_{12} & a_{13} & . &.  & . &a_{1n} \\
a_{21} & a_{22} & a_{23} & . &.  & . &a_{2n} \\
 .& . & .&  .&. & . &.\\
 .& . &  .& . & . & . &. \\
. & . & . & . & . & . & .\\
a_{n1} &a_{n2}  & a_{n3} & . &.  & . & a_{nn}
 
\end{bmatrix}$

Then the Cofactor Matrix C = $\begin{bmatrix}
C_{11} & C_{12} & C_{13} & . &.  & . &C_{1n} \\
C_{21} & C_{22} & C_{23} & . &.  & . &C_{2n} \\
 .& . & .&  .&. & . &.\\
 .& . &  .& . & . & . &. \\
. & . & . & . & . & . & .\\
C_{n1} &C_{n2}  & a_{n3} & . &.  & . & C_{nn}
 
\end{bmatrix}$

Then the adjoint of Matrix A
adj(A) = $\begin{bmatrix}
a_{11} & a_{21} & a_{31} & . &.  & . &a_{n1} \\
a_{12} & a_{22} & a_{32} & . &.  & . &a_{n2} \\
 .& . & .&  .&. & . &.\\
 .& . &  .& . & . & . &. \\
. & . & . & . & . & . & .\\
a_{m1} &a_{m2}  & a_{m3} & . &.  & . & a_{mn}
 
\end{bmatrix}$

Matrix row operations

Two matrices are said to be row equivalent if one can be obtained from the other by a finite sequence of elementary row operations.
Elementary row operations
  1. Interchange two rows.
  2. Multiply a row by a non zero constant.
  3. Add a multiple of a row to another row.

Inverse of a Matrix

The inverse of a square matrix of order n is a square matrix of order n denoted by $A^{-1}$ such that
A$A^{-1}$= I = $A^{-1}$A   where I is the identity matrix of order n.
A is called an invertible matrix if $A^{-1}$ exists.

If A is an invertible Matrix, then
$A^{-1}=\frac{1}{|A|}adj(A)$
 
The inverse is also found using row reductions. The given Square Matrix A of order n is transformed into an Identity Matrix of order n using row operations. The same row operations are done on the Identity matrix and the resulting matrix is $A^{-1}$.