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What are Determinants?

To every square matrix A = [aij] of order n, we can associate a number (real or complex) called determinant of the square matrix A, where aij = (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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