What are Determinants?

To every square matrix A = [a_ij] of order n, we can associate a number (real or complex) called determinant of the square matrix A, where a_ij = (i, j)^th element of A.

Determinant of a matrix of order one

Let A = [a] be the matrix of order 1, then determinant of A is defined to be equal to ‘a’.
For example:  If A = [5] then lAl = 5

Determinant of a matrix of order two

Determinant of a matrix of order three

Determinant of a matrix of order three can be determined by expressing it in terms of second order determinants. This is known as expansion of a determinant along a row or a column. There are six ways of expanding a determinant of order 3 corresponding to each three rows and three columns.  Commonly we use the expansion along R₁ (row 1).

Here, a₁₁ = 3, a₁₂ = -1, a₁₃ = -2
         a₂₁ = 0, a₂₂ = 0, a₂₃ = -1
         a₃₁ = 3, a₃₂ = -5, a₃₃ = 0
According to formula:
a₁₁ (a₂₂ x a₃₃ - a₃₂ x a₂₃) – a₁₂ (a₂₁ x a₃₃ - a₃₁ x a₂₃) + a₁₃ (a₂₁ x a₃₂ - a₃₁ x a₂₂)
Substituting the values in the above formula, we get:
= 3 (0 - 5) – (-1) (0 – (-3)) – 2 (0 – 0) = -15 + 3 – 0 = -12
Try this:
What value of x makes the determinant −4?

                                        

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Reference Links:

• http://en.wikipedia.org/wiki/Determinant
• http://www.britannica.com/EBchecked/topic/561660/square-matrix
• http://www.mathreference.com/la-mpoly,order.html
• http://en.wikipedia.org/wiki/Row_and_column_spaces