DETERMINANTS

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

i) If A = a₁₁ a₁₂a₂₁ a₂₂ is a square matrix of order 2 X 2, then its

determinant is denoted by

|A| or, a₁₁ a₁₂a₂₁ a₂₂ and is defined as a11 a22 – a12 a21.

i.e. |A| = a₁₁ a₁₂a₂₁ a₂₂ = a11 a22 – a12 a21

ii) If A = a₁₁ a₁₂ a₁₃a₂₁ a₂₂ a₂₃a₃₁ a₃₂ a₃

is a square matrix of order 3 X 3,

a₁₁ a₁₂ a₁₃a₂₁ a₂₂ a₂₃a₃₁ a₃₂ a₃

then its determinant is denoted by |A| or, a₁₁ a₁₂ a₁₃a₂₁ a₂₂ a₂₃a₃₁ a₃₂ a₃

and is equal to a11 a22 a33 + a12 a23 a31 + a13 a32 a21 – a11 a23 a32 - a22

a13 a31 – a12 a21 a33

This expression can be arranged in the following form:

a₁₁ a₁₂ a₁₃a₂₁ a₂₂ a₂₃a₃₁ a₃₂ a₃

= (-1)1 + 1 a11 a₂₂ a₂₃a₃₂ a₃ + (-1)1 + 2 a12

a₂₂ a₂₃a₃₂ a₃₃

+ (-1)1 + 3 a13 a₂₁ a₂₂a₃₁ 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 −21 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 −21 2 3

−2 1 −3

D = (-1)1+1 (2) 2 31 −3 + (-1)2+1 (1) 3 −2

1 −3 + (-1)3+1 (-2)3 −21 −3

D = 22 31 −3 -

3 −21 −3 -23 −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.

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|.

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.