Topics covered in this snacksized chapter:
A matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. The individual items in a matrix are called its elements or entries.
A set of mn numbers arranged in the form of an ordered set of m rows and n columns is called m × n matrix (to be read as m by n matrix).
A matrix with m rows and n columns can be written as:
Where represents the element at the intersection of ^{th
} row and ^{th
} column.
A matrix in which the number of rows is equal to the number of columns is called a Square Matrix.
 Thus a matrix A with m rows and n columns will be a square matrix if
m = n.
In a square matrix all those elements for which that is, all those elements which occur in the same row and same column namely are called the diagonal elements.
A square matrix A is said to be a diagonal matrix if all its nondiagonal elements be zero.
A diagonal matrix whose all the diagonal elements are equal, is called a scalar matrix.
A square matrix A all of whose nondiagonal elements are zero and all the main diagonal elements are unity.
In general for a unit matrix,
Zero Matrix or Null Matrix
Any m × n matrix in which all the elements are zero is called a zero matrix or a null matrix of the type m × n and is denoted by
A symmetric matrix is a square matrix that satisfies A^{T
}= A where A^{T
} represents the transpose such that a_{ij
} = a_{ji
}.
It also implies A^{1
} A = I where I is an identity matrix.
Let A be the symmetric matrix in which
Row Matrix is a matrix with only one row.
For example:
[]
A matrix which has only one column is called a Column Matrix.
For example:
Two matrices are said to be equal if they have the same order and their corresponding entries are equal.
For example:
and are equal matrix because both the matrix A and
B
have same order and same corresponding entries.
Matrix addition is commutative:
A + B = B + A
Matrix addition is associative:
A + (B + C) = (A + B) + C
Multiplication of Matrices is distributive with respect to addition of matrices.
A (B + C) = AB + AC
Matrix Multiplication is associative if conformability is assured.
A(BC) = (AB)C
The multiplication of Matrices is not always commutative.
 AB is not always equal to BA.
Multiplication of a Matrix A by a null matrix conformable with A for multiplication is a null matrix.
If AB = 0 then it does not necessarily mean that either A = 0 or B = 0 or both are 0.
Multiplication of Matrix A by a Unit Matrix
Let A be a (m × n) matrix and I be a square unit matrix of order n, so that A and I are conformable for multiplication then
If A be a given matrix of the type m × n then the matrix obtained by changing the rows of A into columns and columns of A into rows is called Transpose of matrix A and is denoted by
As there are rows in therefore there will be columns in and similarly as there are columns in therefore there will be rows in
Then
(A')' = A
(KA)'= KA’, K being a scalar
(A + B)' = A' + B'
(AB)' = B'A'
(ABC)' = C'B'A'
Consider the set of equations
Or
The above sets of equations can be conveniently written in matrix form as shown below:
That is AX = B
Or
That is AX = B
In the above, the matrix is called “coefficient matrix”.
If the above equations have a solution, we say that they are consistent and have either a unique solution or infinite solutions.
In case they do not have any solution, we can say that the system of equations is inconsistent.
have a unique solution.
can be verified by solving them.
 Here the coefficient matrix is
and
A = 6  5 = 11 ≠ 0
 Matrix is nonsingular and its inverse exists.
 In this case, we will have a unique solution. The above equation can be written in matrix form as:
 Where is a nonsingular matrix as: A = 11 and
.
 Multiplying both sides of (1) by A^{1
}, we get:
 Therefore or
Or
Therefore
A determinant is a real number associated with every square matrix.
The determinant of a square matrix is denoted by or.
Determinant of a matrices


A minor for any element is the determinant that results when the row and column of that element are deleted.
For the matrix shown below (Note that R1 is row 1 and C1 is column 1).
 C1
 C2
 C3

R1
 1
 4
 3

R2
 0
 5
 2

R3
 3
 6
 1

Minor for (,, deleted) is:
The matrix of minors is the square matrix where each element is the minor for the number in that position.
Cofactor for any element is either the minor or the opposite of the minor, based on the element's position in the original Determinant.
If the row and column of the element add up to an even number.
 The cofactor is the same as the minor.
If the row and column of the element add up to an odd number.
 The cofactor is the opposite of the minor.
 C1
 C2
 C3

R1
 +
 
 +

R2
 
 +
 

R3
 +
 
 +

The matrix formed by taking the transpose of the cofactor matrix of a given matrix.
The adjoint of matrix is often written as adj A.
It can be found in 6 steps as shown below:
 Find the matrix of minors.
 Turn it into a matrix of cofactors (by changing the sign of the odd elements).
 Find the adjoint by transposing the matrix of cofactors (Columns becomes rows and rows becomes columns).
 Find the determinant of the matrix.
 Divide the adjoint of the matrix by the determinant of the matrix.
 The final result is a matrix inverse.
or
Consider the matrix
The cofactor matrix for A is:
So the adjoint is:
Since det A = 22, we get: