Determinant is a number which is associated with every square matrix. It can be both positive or negative.
The expression a11a22 - a12 a21, associated with matrix A = $\begin{bmatrix} a_{11} &a_{12} \\ a_{21} & a_{22} \end{bmatrix}$
is defined as determinant of A and is denoted by det A or $\left | A \right |$. In linear algebra, we have various ways to define determinant of a matrix. The system of linear equations a11 X+ a12 Y = b11, a21 X + a22 Y = b21 can be written in matrix form as $\begin{bmatrix} a_{11} &a_{12} \\ a_{21} & a_{22} \end{bmatrix}$ $\begin{bmatrix} X\\ Y\end{bmatrix}$ = $\begin{bmatrix} b_{11}\\ b_{21} \end{bmatrix}$ In order to find whether the above system has a unique solution or not, we have to determine the determinant value. If the determinant is
$\neq$ 0, then the system has an unique solution.

## Finding the Determinant of a Matrix

To find the determinant of a matrix, we have to know about minors and cofactors.

Minors:
If A = [aij] be a matrix. Then, minor of an element aij is denoted by Mij and defined as the determinant of the matrix formed by removing or deleting ith row and jth column from the matrix A.

Cofactors: Cofactor of aij is denoted by Cij = $\left (-1 ^{i+j} \right )$ Mij, where i and j are the row number and column number respectively. So, cofactor of any minor, is the minor itself or the opposite of the minor.

To understand more about minors and cofactors, lets take an example:
Let A = $\begin{bmatrix} 2&-5 \\ 6 & -1 \end{bmatrix}$
be a 2x2 order matrix.

Then,
Minor of 2, M11 = -1
Minor of -5, M12 = 6
Minor of 6, M21 = -5
Minor of -1, M22 = 2

Cofactor of 2, C11 = (-1)1+1 x M11 = (-1)2 x (-1) = -1
Cofactor of -5, C12 = (-1)1+2 x M12 = (-1)3 x (6) = -6
Cofactor of 6, C21 = (-1)2+1 x M21 = (-1)3 x (-5) = 5
Cofactor of -1, C22 = (-1)2+2 x M22 = (-1)4 x (2) = 2

There are some ways to find the determinant of the matrix. If the order of a given matrix is 2x2, then its easy to find the value of determinant. If the order of the matrix is more then two, then we use row expansion method or column expansion method.

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