# Determinants

What are determinants?

A determinant is a word related to algebra, and it is found in the most of the calculations in Mathematics. It is implemented in linear equations and also used in the computation of matrices.

As we already know what a matrix is, it is an array of elements or numbers. Hence a determinant of a matrix is defined as the special number or a value of a square matrix. Two vertical lines on either side are used to a denote a determinant as shown below-

|X| or det X-This denotes determinant of matrix X.

Suppose X is a matrix with its elements as shown,

$\begin{bmatrix} 3 & 8\\ 4 & 6 \end{bmatrix}$

The determinant of this matrix is |X| calculated as |X| = 6 * 3 – 8 * 4 = |-14|= +14. The determinant is applied to remove the negativity symbol from the values obtained on the calculation of matrices.

Properties of Determinant

Some of the basic properties of determinants as listed below-

• Transpose of a matrix- |Xt|= |X|- This property shows that the determinant for the matrix X is equal to its transpose.

For example- If |A| = 2 then the transpose matrix for the same will be |At|= -2

• |X|= 0- In this property, the matrix consists of two equal lines having equal values. Hence the resultant matrix would always be zero.
• |A-1| = 1/ A
• |AB|= |A|. |B|- In this property, the determinant of AB is equal to the product of the individual determinants.

How to Solve Determinants

1. To solve 2 * 2 matrix-

Consider a matrix

A= $\begin{bmatrix} a & b\\ c& d \end{bmatrix}$

The determinant for this matrix is calculated as follows-

|A| = ad-bc -value of the determinant.

For example- if

A= $\begin{bmatrix} 4 & 3\\ 6& 8 \end{bmatrix}$

Then |A| = 4 * 8 – 6 * 3= 32- 18= 14.

1. How to solve a 3 * 3 matrix-

Suppose A is a 3 * 3 matrix consisting of elements-

A= $\begin{bmatrix} a & b & c\\ d& e& f\\ g& h& i \end{bmatrix}$

Then the determinant of matrix A is calculated as

|A| = a(ei – fh) – b(di – fg) + c(dh – eg)

For example – consider the matrix

A = $\begin{bmatrix} 6 & 1 & 1\\ 4& -2& 5\\ 2& 8& 7 \end{bmatrix}$

The determinant of A is given by |A| = [6×(-2×7 – 5×8) ]- [1×(4×7 – 5×2)] + [1×(4×8 – -2×2)]
= [6×(-54)] – [1×(18) ]+ [1×(36)]

|A| = |-306| = 306.

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#### Practise This Question

If a polynomial p(x) is of degree “n”, the graph of the polynomial should cut the X axis at “n” points.