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Determinant Formula

Determinant Formula

Determinant in linear algebra is a useful value which is computed from the elements of a square matrix. The determinant of a matrix A is denoted det (A), det A, or |A|. It is a function that has an input accepts

\(\begin{array}{l}n\times n\end{array} \)
matrix and output in a real or a complex number which is called as the determinant of the input matrix. Determinants occur throughout mathematics. For example, a matrix is often used to represent the coefficients in a system of linear equations, and the determinant is used to solve these equations, even though more efficient techniques are actually used, some are determinant-revealing and consist of computationally effective ways of calculating the determinant itself.

If \[\LARGE A=\begin{bmatrix} a & b\\ c & d \end{bmatrix}\]

The Determinant Formula is

 \[\LARGE Determinant=ad-bc\]

If \[\LARGE B=\begin{bmatrix} i & j & k\\ a & b & c\\ x & y & z \end{bmatrix}\]

The Determinant Formula is

\[\LARGE Determinant =i(bz-cy)-j(az-cx)+k(ay-bx)\]

Solved Examples

Question 1: Find out the determinant of 

\(\begin{array}{l}\begin{bmatrix} 4 & 2\\ 7 & 8 \end{bmatrix}\end{array} \)

Solution:

Given 2
\(\begin{array}{l}\times\end{array} \)
 2 matrix is,

A =

\(\begin{array}{l}\begin{bmatrix} 4 & 2\\ 7 & 8 \end{bmatrix}\end{array} \)

Determinant is calculated as,

ad – bc = (4

\(\begin{array}{l}\times\end{array} \)
 8) – (2
\(\begin{array}{l}\times\end{array} \)
7) = 32 – 14 = 18

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