Write the Minors and Cofactors of the elements of the given determinant.
$\left|\begin{array}{ccc}1& 0& 4\\ 3& 5& -1\\ 0& 1& 2\end{array}\right|$

(i) $\left|\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right|=a_{11}{a}_{22}-a_{12}{a}_{21}$.

(ii) a11a12a13a21a22a23a31a32a33=-11+1a11a22a23a32a33+-11+2a12a21a23a31a33+-11+3a13a21a22a31a32

(iii) $\left|\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right|=a_{11}{a}_{22}{a}_{33} + a_{12}{a}_{23}{a}_{31}+a_{13}{a}_{32}{a}_{21}- a_{11}{a}_{23}{a}_{32}–a_{12}{a}_{21}{a}_{33}–a_{22}{a}_{13}{a}_{31}$

(iv) A square matrix with its determinant is zero is called as a singular matrix.

2. Minor of a matrix:

Let $A=\left[a_{ij}\right]$ be a square matrix of order $n$ then the minor $M_{ij}$ of element $a_{ij}$ in $A$ is the determinant of the sub-matrix of order $\left(n-1\right)$ obtained by leaving ${\mathrm{i}}^{\mathrm{th}}$ row and ${\mathrm{j}}^{\mathrm{th}}$ column of $A$.

For example, if A=123-32-12-43, then $M_{11}=\left|\begin{array}{cc}2& -1\\ -4& 3\end{array}\right|, M_{12}=\left|\begin{array}{cc}-3& -1\\ 2& 3\end{array}\right|$

3. Cofactor of a matrix:

The cofactor $C_{ij}$ of $a_{ij}$ in $A={\left[a_{ij}\right]}_{n\times n}$ is equal to ${\left(-1\right)}^{i+j}{M}_{ij}.$

For example, if A=123-32-12-43, then $C_{11}={\left(-1\right)}^{1+1}{M}_{11}=M_{11}=2$.

4. Properties of determinants:

(i) Sum of the product of elements of any row (column) with their cofactors of a matrix is always equal to its determinant i.e. ∑j=1naijCij=Aand ∑i=1naijCij=A.

(ii) $\left|A\right|=\left|A^T\right|$ i.e., the value of a determinant remains unchanged if its rows and columns are interchanged.

(iii) Let $A=\left[a_{ij}\right]$ be a square matrix of order $n\left(\ge 2\right)$ and let $B$ be a matrix obtained from $A$ by interchanging any two rows (columns) of $A$, then $\left|B\right|=-\left|A\right|.$

(iv) If any two rows or columns of a determinant are identical, then its value is zero.

(v) Let $A=\left[a_{ij}\right]$ be a square matrix of order $n,$ and let $B$ be the matrix obtained from $A$ by multiplying each element of a row (column) of $A$ by a scalar $k,$ then $\left|B\right|=k\left|A\right|.$

(vi) If $A=\left[a_{ij}\right]$ be a square matrix of order $n$, then $\left|kA\right|=k^n\left|A\right|.$

(vii) If each element of a row (column) of a determinant is expressed as a sum of two or more terms, then the determinant can be expressed as the sum of two or more determinants.

(viii) If each element of a row (column) of a determinant is multiplied by the same constant and then added to the corresponding elements of some other row (column), then the value of the determinant remains same.

(ix) If each element of a row (column) of a determinant is zero, then its value is zero.

(x) If $A=\left[a_{ij}\right]$ is a diagonal matrix of order $n\left(\ge 2\right),$ then |A|=a11.a22.a33…ann_.

(xi) If $A$ and $B$ are square matrices of the same order, then $\left|AB\right|=\left|A\right|\left|B\right|.$

(xii) If $A=\left[a_{ij}\right]$ is a triangular matrix of order $n,$ then |A|=a11.a22.a33…ann_.

5. Application of determinants:

(i) Area of a triangle with vertices $\left(x_1,y_1\right),\left(x_2,y_2\right)$ and $\left(x_3,y_3\right)$ is given by $\Delta =\frac{1}{2}\left|\left|\begin{array}{ccc}{x}_1& y_1& 1\\ x_2& y_2& 1\\ x_3& y_3& 1\end{array}\right|\right|$.

(ii) Cramer’s Rule: 

(a) The solution of the system of linear equations $a_{1}x+b_{1}y+c_{1}z=d_1$, $a_{2}x+b_{2}y+c_{2}z=d_2$ and $a_{3}x+b_{3}y+c_{3}z=d_3$ is given by $x=\frac{D_1}{D},y=\frac{D_2}{D}$ and z=D3D, where $D=\left|\begin{array}{ccc}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right|, D_1=\left|\begin{array}{ccc}{d}_1& b_1& c_1\\ d_2& b_2& c_2\\ d_3& b_3& c_3\end{array}\right|, D_2=\left|\begin{array}{ccc}{a}_1& d_1& c_1\\ a_2& d_2& c_2\\ a_3& d_3& c_3\end{array}\right|$and D3=a1b1d1a2b2d2a3b3d3, provided that $D\ne 0.$

(b) If $D\ne 0,$ then the given system of equations is consistent and has a unique solution given by $x=\frac{D_1}{D},y=\frac{D_2}{D}$ andz=D3D.

(c) If $D=0$ and $D_1=D_2=D_3=0,$ then the given system of equations is consistent with infinitely many solutions.

(d) If $D=0$ and at least one of the determinants $D_1,D_2,D_3$ is non-zero, then the given system of equations is inconsistent.

Note: For a system of linear equations with two variables, determinants become $2\times 2$ matrices.