# 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 R_{1} (row 1).

Here, a_{11} = 3, a_{12} = -1, a_{13} = -2

a_{21} = 0, a_{22} = 0, a_{23} = -1

a_{31} = 3, a_{32} = -5, a_{33} = 0

According to formula:

a_{11} (a_{22} x a_{33} - a_{32} x a_{23}) – a_{12} (a_{21} x a_{33} - a_{31} x a_{23}) + a_{13} (a_{21} x a_{32} - a_{31} x a_{22})

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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