1. Every square matrix can be associated to an expression or a number which is known as its determinant.

1. DETERMINANTS
1. Every square matrix can be associated to an expression or a
number which is known as its determinant.
i) If A =
𝑎₁₁ 𝑎₁₂
𝑎₂₁ 𝑎₂₂ is a square matrix of order 2 X 2, then its
determinant is denoted by
|A| or,
𝑎₁₁ 𝑎₁₂
𝑎₂₁ 𝑎₂₂ and is defined as a11 a22 – a12 a21.
i.e. |A| =
𝑎₁₁ 𝑎₁₂
𝑎₂₁ 𝑎₂₂ = a11 a22 – a12 a21
ii) If A =
𝑎₁₁ 𝑎₁₂ 𝑎₁₃
𝑎₂₁ 𝑎₂₂ 𝑎₂₃
𝑎₃₁ 𝑎₃₂ 𝑎₃
is a square matrix of order 3 X 3,
𝑎₁₁ 𝑎₁₂ 𝑎₁₃
𝑎₂₁ 𝑎₂₂ 𝑎₂₃
𝑎₃₁ 𝑎₃₂ 𝑎₃
then its determinant is denoted by |A| or,
𝑎₁₁ 𝑎₁₂ 𝑎₁₃
𝑎₂₁ 𝑎₂₂ 𝑎₂₃
𝑎₃₁ 𝑎₃₂ 𝑎₃
and is equal to a11 a22 a33 + a12 a23 a31 + a13 a32 a21 – a11 a23 a32 -
a22 a13 a31 – a12 a21 a33

2. This expression can be arranged in the following form:
𝑎₁₁ 𝑎₁₂ 𝑎₁₃
𝑎₂₁ 𝑎₂₂ 𝑎₂₃
𝑎₃₁ 𝑎₃₂ 𝑎₃
= (-1)1 + 1 a11
𝑎₂₂ 𝑎₂₃
𝑎₃₂ 𝑎₃ + (-1)1 + 2 a12
𝑎₂₂ 𝑎₂₃
𝑎₃₂ 𝑎₃₃
+ (-1)1 + 3 a13
𝑎₂₁ 𝑎₂₂
𝑎₃₁ 𝑎₃₂
This is known as the expansion of |A| along first row.
In fact, |A| can be expanded along any of its rows or columns. In order
to expand |A| along any row or column, we multiply
Example 1: - Evaluate the determinant
D =
2 3 −2
1 2 3
−2 1 −3
by expanding it along first column.
SOLUTION: By using the definition, of expansion along first column, we
obtain
D =
2 3 −2
1 2 3
−2 1 −3

3. D = (-1)1+1 (2)
2 3
1 −3
+ (-1)2+1 (1)
3 −2
1 −3
+ (-1)3+1 (-2)
3 −2
1 −3
D = 2
2 3
1 −3
-
3 −2
1 −3
-2
3 −2
1 −3
D = 2 (-6-3) – (-9+2) -2(9+4) = -18 +7-26 = -37.
NOTE 1: Only square matrices have their determinants. The matrices
which are not square do not have determinants.
NOTE 2: The determinant of a square matrix of order 3 can be
expressed along any row or column.
NOTE 3: If a row or a column of a determinant consists of all zeros, then
the value of the determinant is zero.
There are three rows and three columns in a square matrix of order 3.
PROPERTIES OF DETERMINANTS
We have defined the determinants of a square matrix of order 4 or less.
In fact, these definitions are consequences of the general definition of
the determinant of a square matrix of any order which needs so many
advanced concepts. These concepts are beyond the scope of this book.
Using the said definition and some other advanced concepts we can
prove the following properties. But, the concepts used in the definition
itself are very advanced. Therefore we mention and verify them for a
determinant of a square matrix of order 3.

4. Property 1: let A = [aij] be a square matrix of order n, then the sum of
the product of elements of any row(column) with their cofactors is
always equal to |A| or, det (A).
Property 2: let A = [aij] be a square matrix of order n, then the sum of
the product of elements of any row(column) with the cofactors of the
corresponding elements of some other row (column) is zero.
Property 3: Let A = [aij] be a square matrix of order n, then |A| = |AT|.
Property 4: let A = [aij] be a square matrix of order n(≥2) and let B be a
matrix obtained from A by interchanging any two rows(columns) of A,
then |B| = -|A|.
Conventionally this property is also stated as:
1. If any two rows (columns) of a determinant are interchanged, then
the value of the determinant changes by minus sign only.
Property 5: if any two rows (columns) of a square matrix A = [aij] of
order n (>2) are identical, then its determinant is zero i.e. |A| = 0.
Property 6: Let A = [aij] be a square matrix of order n, and let B be
the matrix obtained from A by multiplying each element of a row
(column) of A by a scalar k, then |B| = k |A|.

5. Property 7: Let A square matrix such that each element of row
(column) of A is expressed as the sum of two or more terms. Then,
the determinant of A can be expressed as the sum of the
determinants of two or more matrices of the same order.