Determinants

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Let A = [a_ij] be a square matrix. We can associate with the square matrix A, a determinant which is formed by exactly the same array of elements of the matrix A. A determinant formed by the same array of elements of the square matrix A is called the determinant of the square matrix A and is denoted by the symbol det.A or |A|.

It should be remembered that the determinant of a square matrix will be a scalar quantity. i.e., with a determinant we associate a definite value, whereas a matrix is essentially an arrangement of numbers and has no value.

For example,

To find the determinant of a matrix of order 3 (or more) we need the following definitions.

Minors

Let |aij| be a determinant of order n.

The determinant obtained by deleting the i^th row and j^th column in which the element a_ij lies is called the minor of element a_ij and is denoted by M_ij.

Co-factors

The co-factor of the element a_ij is (-1)^i+j times its minor a_ij. We shall denote the cofactor of an element by the corresponding capital letter.

Cofactor of a_ij = A_ij = (-1)^i+j M_ij

Consider the determinant

The minor of a₁₁ can be obtained by deleting the first row and first column of D. The Determinant, so obtained after deletion is the minor of a₁₁ and is denoted by M₁₁.

The minor of a₁₂ is obtained as follows

Note that the position is 1^st row and 2^nd column. Delete the 1^st row and 2^nd column, the determinant so obtained is the minor of a₁₂

That is minor of a₁₂

The co-factor of a₁₂ = A₁₂ = (-1)¹⁺² M₁₂

= - (a₂₁a₃₃ - a₃₁a₂₃)

Similarly the minor of a₁₃

= a₂₁a₃₂ - a₃₁a₂₂

Co-factor of a₁₃ = A₁₃ = (-1)¹⁺³M₁₃

= (a₂₁a₃₂ - a₃₁a₂₂)

The value of the determinant D is given by a₁₁A₁₁ + a₁₂A₁₂ + a₁₃A₁₃

This is first row expansion of the determinant. Similarly, we can expand the determinant along any row or any column.

In general,

where i = 1, 2 and 3

or

Where j = 1, 2 and 3.

Properties of Determinants

• If the rows and columns of a determinant are inter-changed, the value remains unaltered.

• If any two rows (columns) of a determinant are identical, its value of the determinant is zero.

• If any two rows (columns) of a determinant are interchanged, the value of the determinant is (-1) times the original determinant.

• If all the elements of one row (column) of a determinant is multiplied by k, the value of the new determinant is k times the original determinant.

• If to any row or column of a determinant, a multiple of another row or column is added, the value of the determinant remains the same.

• If some or all the elements of a row (or column) of a determinant are expressed as sum of two (or more) terms, then the determinant can be expressed as sum of two or more determinants. Thus

• The sum of the products of the elements of any row (column) with their corresponding cofactors is equal to the value of the determinant.

• The sum of the products of the elements of any row (column) and the cofactors of the corresponding elements of any other row (column) is zero.

Example: For a matrix of order 3,

a₁₁A₂₁ + a₁₂A₂₂ + A₁₃A₂₃ = 0