\displaystyle \textbf{1. }\ \text{Every square matrix can be associated to an expression or a number which is known} \\ \text{as its determinant.}

\displaystyle \textbf{(i) }\ \text{If }A=\begin{bmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{bmatrix}\text{ is a square matrix of order }2\times2,\text{ then its determinant} \\ \text{is denoted by} |A|\text{ or }\begin{vmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{vmatrix}\text{ and is defined as }a_{11}a_{22}-a_{12}a_{21}.
\displaystyle \text{i.e. }|A|=\begin{vmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{vmatrix}=a_{11}a_{22}-a_{12}a_{21}.

\displaystyle \textbf{(ii) }\ \text{If }A=\begin{bmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{bmatrix}\text{ is a square matrix of order }3\times3,\text{ then its determinant} \\ \text{is denoted by} |A|\text{ or }\begin{vmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{vmatrix}\text{ and is equal to } \\ 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_{22}a_{13}a_{31}-a_{12}a_{21}a_{33}.
\displaystyle \text{This expression can be arranged in the following form:}
\displaystyle \begin{vmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{vmatrix}=(-1)^{1+1}a_{11}\begin{vmatrix}a_{22}&a_{23}\\a_{32}&a_{33}\end{vmatrix}+(-1)^{1+2}a_{12}\begin{vmatrix}a_{21}&a_{23}\\a_{31}&a_{33}\end{vmatrix}+(-1)^{1+3}a_{13}\begin{vmatrix}a_{21}&a_{22}\\a_{31}&a_{32}\end{vmatrix}.
\displaystyle \text{This is known as the expansion of }|A|\text{ along first row.}
\displaystyle \text{In fact, }|A|\text{ can be expanded along any of its rows or columns. In order to expand } \\ |A|\text{ along any row or column, we multiply each element }a_{ij}\text{ of }i\text{th row (say) with } \\ (-1)^{i+j}\text{ times the determinant of the submatrix obtained by leaving the row and} \\ \text{column passing through the element and then they are added.}
\displaystyle \text{Similarly, we can find the value of the determinant of square matrices of order }4\text{ or more.}

\displaystyle \textbf{2. }\ \text{A square matrix is a singular matrix if its determinant is zero. Otherwise, it is a} \\ \text{non-singular matrix.}

\displaystyle \textbf{3. }\ \textbf{(i) }\ \text{Let }A=[a_{ij}]\text{ be a square matrix of order }n.\text{ Then the minor }M_{ij}\text{ or }a_{ij}\text{ in } \\ A\text{ is the determinant of the sub-matrix of order }(n-1)\text{ obtained by leaving }i\text{th row} \\ \text{and }j\text{th column of }A.
\displaystyle \text{For example, if }A=\begin{bmatrix}1&2&3\\-3&2&-1\\2&-4&3\end{bmatrix},\text{ then }M_{11}=\begin{vmatrix}2&-1\\-4&3\end{vmatrix}=2,\ M_{12}=\begin{vmatrix}-3&-1\\2&3\end{vmatrix}=-7\text{ and so on.}

\displaystyle \textbf{(ii) }\ \text{The cofactor }C_{ij}\text{ of }a_{ij}\text{ in }A=[a_{ij}]_{n\times n}\text{ is equal to }(-1)^{i+j}\text{ times }M_{ij}.
\displaystyle \text{For example, if }A=\begin{bmatrix}1&2&3\\-3&2&-1\\2&-4&3\end{bmatrix},\text{ then }C_{11}=(-1)^{1+1}M_{11}=M_{11}=2\text{ and }C_{12}=(-1)^{1+2}M_{12}=-M_{12}=7\text{ and so on.}

\displaystyle \textbf{4. }\ \text{Following are some important properties of determinants:}

\displaystyle \textbf{(i) }\ \text{Let }A=[a_{ij}]\text{ be a square matrix of order }n,\text{ then the sum of the product of elements of} \\ \text{any row (column) with their cofactors is always equal to }|A|\text{ or, }\det(A).
\displaystyle \text{i.e. }\sum_{j=1}^{n}a_{ij}C_{ij}=|A|\text{ and }\sum_{i=1}^{n}a_{ij}C_{ij}=|A|.

\displaystyle \textbf{(ii) }\ \text{Let }A=[a_{ij}]\text{ be a square matrix of order }n,\text{ then the sum of the product of elements of any} \\ \text{row (column) with the cofactors of the corresponding elements of some other row (column) is zero.}
\displaystyle \text{i.e. }\sum_{j=1}^{n}a_{ij}C_{kj}=0\text{ and }\sum_{i=1}^{n}a_{ij}C_{ik}=0.

\displaystyle \textbf{(iii) }\ \text{Let }A=[a_{ij}]\text{ be a square matrix of order }n,\text{ then }|A|=|A^{T}|.
\displaystyle \text{By the abuse of language this property is also stated as follows:}
\displaystyle \text{The value of a determinant remains unchanged if its rows and columns are interchanged.}

\displaystyle \textbf{(iv) }\ \text{Let }A=[a_{ij}]\text{ be a square matrix of order }n(\ge2)\text{ and let }B\text{ be a matrix obtained from }A\text{ by} \\ \text{interchanging any two rows (columns) of }A,\text{ then }|B|=-|A|.
\displaystyle \text{Conventionally this property is also stated as:}
\displaystyle \text{If any two rows (columns) of a determinant are interchanged, then the value of the} \\ \text{determinant changes by} \\ \text{minus sign only.}

\displaystyle \textbf{(v) }\ \text{If any two rows (columns) of a square matrix }A=[a_{ij}]\text{ of order }n(\ge2)\text{ are identical,} \\ \text{then its determinant is zero i.e. }|A|=0.
\displaystyle \text{Conventionally this property is stated as:}
\displaystyle \text{If any two rows or columns of a determinant are identical, then its value is zero.}

\displaystyle \textbf{(vi) }\ \text{Let }A=[a_{ij}]\text{ be a square matrix of order }n,\text{ and let }B\text{ be the matrix obtained from } \\ A\text{ by multiplying each element of a row (column) of }A\text{ by a scalar }k,\text{ then }|B|=k|A|.
\displaystyle \text{Conventionally this property is also stated as:}
\displaystyle \text{If each element of a row (column) of a determinant is multiplied by a constant }k, \\ \text{ then the value of the new determinant is }k\text{ times the value of the original determinant.}
\displaystyle \text{If }A=[a_{ij}]\text{ be a square matrix of order }n,\text{ then }|kA|=k^{n}|A|.

\displaystyle \textbf{(vii) }\ \text{Let }A\text{ be a square matrix such that each element of a row (column) of }A\text{ is expressed} \\ \text{as the sum of two or more terms. Then the determinant of }A\text{ can be expressed as the sum} \\ \text{of the determinants of two or more matrices of the same order.}
\displaystyle \text{Conventionally this property is also stated as:}
\displaystyle \text{If each element of a row (column) of a determinant is expressed as a sum of two or more} \\ \text{terms, then the determinant can be expressed as the sum of two or more determinants.}

\displaystyle \textbf{(viii) }\ \text{Let }A\text{ be a square matrix and }B\text{ be a matrix obtained from }A\text{ by adding to} \\ \text{a row (column) of }A\text{ a scalar multiple of another row (column) of }A,\text{ then }|B|=|A|.
\displaystyle \text{This property is conventionally stated as:}
\displaystyle \text{If each element of a row (column) of a determinant is multiplied by the same constant} \\ \text{and then added to the corresponding elements of some other row (column), then the} \\ \text{value of the determinant remains same.}

\displaystyle \textbf{(ix) }\ \text{Let }A\text{ be a square matrix of order }n(\ge2)\text{ such that each element in a row} \\ \text{(column) of }A\text{ is zero, then }|A|=0.
\displaystyle \text{Conventionally this property is also stated as:}
\displaystyle \text{If each element of a row (column) of a determinant is zero, then its value is zero.}

\displaystyle \textbf{(x) }\ \text{If }A=[a_{ij}]\text{ is a diagonal matrix of order }n(\ge2),\text{ then }|A|=a_{11}.a_{22}.a_{33}\ldots a_{nn}.

\displaystyle \textbf{(xi) }\ \text{If }A\text{ and }B\text{ are square matrices of the same order, then }|AB|=|A||B|.

\displaystyle \textbf{(xii) }\ \text{If }A=[a_{ij}]\text{ is a triangular matrix of order }n,\text{ then }|A|=a_{11}.a_{22}.a_{33}\ldots a_{nn}.

\displaystyle \textbf{5. }\ \text{Area of a triangle with vertices }(x_{1},y_{1}),\ (x_{2},y_{2})\text{ and }(x_{3},y_{3})\text{ is given by}
\displaystyle \Delta=\frac{1}{2}\begin{vmatrix}x_{1}&y_{1}&1\\x_{2}&y_{2}&1\\x_{3}&y_{3}&1\end{vmatrix}.

\displaystyle \textbf{6. }\ (i)\ \text{If }A\text{ is a skew-symmetric matrix of odd order, then }|A|=0.

\displaystyle \textbf{(ii) }\ \text{The determinant of a skew-symmetric matrix of even order is a perfect square.}

\displaystyle \textbf{7. }\ \text{Consider a system of simultaneous linear equations given by}
\displaystyle a_{1}x+b_{1}y+c_{1}z=d_{1}
\displaystyle a_{2}x+b_{2}y+c_{2}z=d_{2}
\displaystyle a_{3}x+b_{3}y+c_{3}z=d_{3}

\displaystyle \text{A set of values of the variables }x,y,z\text{ which simultaneously satisfy these three equations is} \\ \text{called a solution.}

\displaystyle \text{A system of linear equations may have a unique solution, or many solutions, or no solution} \\ \text{at all. If it has a solution (whether unique or not) the system is said to be consistent. If it has} \\ \text{no solution, it is called an inconsistent system.}

\displaystyle \text{If }d_{1}=d_{2}=d_{3}=0\text{ in (i), then the system of equations is said to be a homogeneous} \\ \text{system. Otherwise it is called a non-homogeneous system of equations.}

\displaystyle \textbf{(i) }\ (\text{Cramer's rule})\ \text{The solution of the system of simultaneous linear equations}
\displaystyle a_{1}x+b_{1}y=c_{1}
\displaystyle a_{2}x+b_{2}y=c_{2}
\displaystyle \text{is given by }x=\frac{D_{1}}{D},\ y=\frac{D_{2}}{D},\ \text{where}
\displaystyle D=\begin{vmatrix}a_{1}&b_{1}\\a_{2}&b_{2}\end{vmatrix},\ D_{1}=\begin{vmatrix}c_{1}&b_{1}\\c_{2}&b_{2}\end{vmatrix}\text{ and }D_{2}=\begin{vmatrix}a_{1}&c_{1}\\a_{2}&c_{2}\end{vmatrix}\text{ provided that }D\ne0.

\displaystyle \textbf{(ii) }\ (\text{Cramer's Rule})\ \text{The solution of the system of linear equations}
\displaystyle a_{1}x+b_{1}y+c_{1}z=d_{1}
\displaystyle a_{2}x+b_{2}y+c_{2}z=d_{2}
\displaystyle a_{3}x+b_{3}y+c_{3}z=d_{3}
\displaystyle \text{is given by }x=\frac{D_{1}}{D},\ y=\frac{D_{2}}{D}\text{ and }z=\frac{D_{3}}{D},\ \text{where}
\displaystyle D=\begin{vmatrix}a_{1}&b_{1}&c_{1}\\a_{2}&b_{2}&c_{2}\\a_{3}&b_{3}&c_{3}\end{vmatrix},\ D_{1}=\begin{vmatrix}d_{1}&b_{1}&c_{1}\\d_{2}&b_{2}&c_{2}\\d_{3}&b_{3}&c_{3}\end{vmatrix},\ D_{2}=\begin{vmatrix}a_{1}&d_{1}&c_{1}\\a_{2}&d_{2}&c_{2}\\a_{3}&d_{3}&c_{3}\end{vmatrix}\text{ and }D_{3}=\begin{vmatrix}a_{1}&b_{1}&d_{1}\\a_{2}&b_{2}&d_{2}\\a_{3}&b_{3}&d_{3}\end{vmatrix} \\ \text{ provided that }D\ne0.

\displaystyle \textbf{8. }\ (a)\ \text{For a system of 2 simultaneous linear equations with 2 unknowns:}

\displaystyle \textbf{(i) }\ \text{If }D\ne0,\text{ then the given system of equations is consistent and has a unique solution} \\ \text{given by }x=\frac{D_{1}}{D},\ y=\frac{D_{2}}{D}.

\displaystyle \textbf{(ii) }\ \text{If }D=0\text{ and }D_{1}=D_{2}=0,\text{ then the system is consistent and has infinitely} \\ \text{many solutions.}

\displaystyle \textbf{(iii) }\ \text{If }D=0\text{ and one of }D_{1}\text{ and }D_{2}\text{ is non-zero, then the system} \\ \text{is inconsistent.}

\displaystyle (b)\ \text{For a system of 3 simultaneous linear equations in three unknowns:}

\displaystyle \textbf{(i) }\ \text{If }D\ne0,\text{ then the given system of equations is consistent and has a unique} \\ \text{solution given by }x=\frac{D_{1}}{D},\ y=\frac{D_{2}}{D}\text{ and }z=\frac{D_{3}}{D}.

\displaystyle \textbf{(ii) }\ \text{If }D=0\text{ and }D_{1}=D_{2}=D_{3}=0,\text{ then the given system of equations is} \\ \text{consistent with infinitely many solutions.}

\displaystyle \textbf{(iii) }\ \text{If }D=0\text{ and at least one of the determinants }D_{1},D_{2},D_{3}\text{ is non-zero, then} \\ \text{the given system of equations is inconsistent.}

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