`color{green} ✍️` 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 :

`A = [a ]` be the matrix of order 1, then determinant of A is defined to be equal to `color{orange}{|A| = a}`

Determinant of a matrix of order `2 × 2 : `

`=>` If `A = [(a_(11), a_(12)),(a_(21), a_(22))]` be a matrix of order 2 × 2,

then the determinant of A is defined as `color{orange}{|A| = |(a_(11), a_(12)),(a_(21), a_(22))|}`

Determinant of a matrix of order `3 × 3 : `

`color{green} ✍️` If `A = [(a_(11), a_(12),a_13),(a_(21), a_(22),a_23),(a_31,a_32,a_33)]` be a matrix of order `3 × 3, `

then the determinant of `A` is defined as `color{orange} |A| = |(a_(11), a_(12),a_13),(a_(21), a_(22),a_23),(a_31,a_32,a_33)| `

Determinant of a matrix of order one :

`A = [a ]` be the matrix of order 1, then determinant of A is defined to be equal to `color{orange}{|A| = a}`

Determinant of a matrix of order `2 × 2 : `

`=>` If `A = [(a_(11), a_(12)),(a_(21), a_(22))]` be a matrix of order 2 × 2,

then the determinant of A is defined as `color{orange}{|A| = |(a_(11), a_(12)),(a_(21), a_(22))|}`

Determinant of a matrix of order `3 × 3 : `

`color{green} ✍️` If `A = [(a_(11), a_(12),a_13),(a_(21), a_(22),a_23),(a_31,a_32,a_33)]` be a matrix of order `3 × 3, `

then the determinant of `A` is defined as `color{orange} |A| = |(a_(11), a_(12),a_13),(a_(21), a_(22),a_23),(a_31,a_32,a_33)| `

`color{green} ✍️` The area of triangle whose vertices are `(x_1,y_1)` , `(x_2,y_2)` and `(x_3,y_3)`

`= |1/2[x_1(y_2 -y_3) + x_2(y_3-y_1) + x_3(y_1-y_2)]|`

● Now this expression can be written in the form of a determinant as

`color{red}{"Area =" 1/2 |(x_1,y_1,1),(x_2,y_2,1),(x_3,y_3,1) |}`

`= |1/2[x_1(y_2 -y_3) + x_2(y_3-y_1) + x_3(y_1-y_2)]|`

● Now this expression can be written in the form of a determinant as

`color{red}{"Area =" 1/2 |(x_1,y_1,1),(x_2,y_2,1),(x_3,y_3,1) |}`

`color{green} ✍️` Minor of an element `a_(ij)` of a determinant is the determinant obtained by deleting its `i^(th)` row and `j^(th)` column in which element `a_(ij)` lies. Minor of an element `a_(ij)` is denoted by `M_(ij)` .

`color{green} ✍️` Cofactor of an element `a_(ij) `, denoted by `A_(ij)` is defined by

`=>color{red} {"cofactor of element " a_(ij) "is define as " A_(ij) = (-1)^(i+j)M_(ij)}`

`=>color{red} {"cofactor of element " a_(ij) "is define as " A_(ij) = (-1)^(i+j)M_(ij)}`

Theorem 1 : Sum of Product of any element with their corresponding Cofactor is equal to the value of Determinants.

Theorem 2 : If elements of a row (or column) are multiplied with cofactors of any other row (or column), then their sum is zero

Theorem 2 : If elements of a row (or column) are multiplied with cofactors of any other row (or column), then their sum is zero

`\color{green} ✍️` The adjoint of a square matrix `A = [a_(ij ) ]_(n × n)` is defined as the transpose of the matrix `[A_(ij) ]_(n × n)`,

` => color {red} {A_(ij) " is the cofactor of the element" a_(ij)}` . Adjoint of the matrix `A` is denoted by `adj A.`

Theorem 1 : `color{red}{A(adj A) = (adj A) A = | A | I}`

Theorem 2 : `color{red}{"If A is a square matrix of order n, then" |adj(A)| = |A|^(n – 1)}`.

` => color {red} {A_(ij) " is the cofactor of the element" a_(ij)}` . Adjoint of the matrix `A` is denoted by `adj A.`

Theorem 1 : `color{red}{A(adj A) = (adj A) A = | A | I}`

Theorem 2 : `color{red}{"If A is a square matrix of order n, then" |adj(A)| = |A|^(n – 1)}`.

`"Singular Matrix :"`

`color{red}{"A square matrix A is said to be singular if" | A | = 0.}`

`"NonSingular Matrix : "`

`color{red}{"A square matrix A is said to be non-singular if "| A | ≠ 0}`

`color{red}{"A square matrix A is said to be singular if" | A | = 0.}`

`"NonSingular Matrix : "`

`color{red}{"A square matrix A is said to be non-singular if "| A | ≠ 0}`

`color{green}{A^(–1) = 1/(|A|) adj A}`

So `color{blue}{A^(-1) "exist iff" |A| ne 0}`

So `color{blue}{A^(-1) "exist iff" |A| ne 0}`