Algebra Of Determinants

Determinant

Every square matrix A is associated with a number, called its determinant and it is denoted by det. $\left(A\right)or\left|A\right|.$

Only square matrices have determinant . The matrices which are not square do not have determinants

1. First Order Determinant

If $A=\left[a\right],$then det $\left(A\right)=\left|A\right|=a$

2. Second Order Determinant

If $A=\left[\begin{array}{c}{a}_{11}{a}_{12}\\ a_{21}{a}_{22}\end{array}\right]$, then

$\left|A\right|=a_{11}{a}_{22}-a_{21}{a}_{12}$

3. Third Order Determinant

If $\left[\begin{array}{c}{a}_{11}{a}_{12}{a}_{13}\\ a_{21}{a}_{22}{a}_{23}\\ a_{32}{a}_{32}{a}_{33}\end{array}\right],$ then

$\left|A\right|=a_{11}\left|\begin{array}{c}{a}_{22}{a}_{23}\\ a_{32}{a}_{33}\end{array}\right|-a_{12}\left|\begin{array}{c}{a}_{21}{a}_{23}\\ a_{31}{a}_{33}\end{array}\right|+a_{13}\left|\begin{array}{c}{a}_{21}{a}_{22}\\ a_{31}{a}_{32}\end{array}\right|$

$or\left|A\right|=a_{11}\left(a_{22}{a}_{33}-a_{32}{a}_{23}\right)-a_{12}\left(a_{21}{a}_{33}-a_{31}{a}_{23}\right)+a_{13}\left(a_{21}{a}_{32}-a_{22}{a}_{31}\right)$

Note

• Expanding a determinant along any row or column gives same value.
• For easier calculations, we will expand the determinant along that row or column which contains maximum number of zeros.
• While expanding, instead of multiplying by ${\left(-1\right)}^{i+j}$ , we can multiply by $+1$ or $-1$ according as $\left(i+j\right)$ is even or odd.

Evaluation of Determinant of Square Matrix of Order 3 by Sarrus Rule

If $\left[\begin{array}{c}{a}_{11}{a}_{12}{a}_{13}\\ a_{21}{a}_{22}{a}_{23}\\ a_{31}{a}_{32}{a}_{33}\end{array}\right],$then determinant can be formed can be formed by enlarging

The matrix by adjoining the first two columns on the right and draw lines as show below parallel and perpendicular to the diagonal.

The value of the determinant, thus will be the sum of the product of element in line parallel to the diagonal minus the sum of the product of elements in line perpendicular to the line segment. Thus

$\Delta =a_{11}{a}_{22}{a}_{33}+a_{12}{a}_{23}{a}_{31}+a_{13}{a}_{21}{a}_{32}-a_{13}{a}_{22}{a}_{31}-a_{11}{a}_{23}{a}_{32}-a_{12}{a}_{21}{a}_{33}.$

Note This method doesn't work for determinants of order greater than 3.

Properties of Derterminants

1. The value of the determinant remains unchanged, if rows unchanged, if rows are changed into columns and columns are changed into rows

e.g. $\left|A^{\prime }\right|=\left|A\right|$

2. If $A={\left[a_{ij}\right]}_{n\times n^{\prime }}n>1$and $B$ be the matrix obtained form $A$by interchanging two of its rows or columns, then

det$\left(B\right)$$=-\det \left(A\right)$

3. If two rows (or columns) of a square matrix $A$are proportional, then $\left|A\right|=0$.

4. $\left|B\right|=k\left|A\right|,$ where $B$ is the matrix obtained form $A$by multiplying one row (or column) of $A$,by .$k$

5. $\left|kA\right|=k^n\left|A\right|,$where $A$ is a matrix of order $n\times n.$

6. If each element of a row (or column) of a determinant is the sum the sum of two or more determinants. e.g.

$\left| \,\begin{matrix} {{a}_{1}}+{{a}_{2}} & b & c \\ {{p}_{1}}+{{p}_{2}} & q & r \\ {{u}_{1}}+{{u}_{2}} & v & w\, \\\end{matrix}\, \right|=\left| \,\begin{matrix} {{a}_{1}} & b & c \\ {{p}_{1}} & q & r \\ {{u}_{1}} & v & w\, \\\end{matrix} \right|+\left| \,\begin{matrix} {{a}_{2}} & b & c \\ {{p}_{2}} & q & r \\ {{u}_{2}} & v & w\, \\\end{matrix} \right|$

7. If the same multiple of the elements of any row (or column) of a determinant are added to the corresponding elements of any other row (or column), then the value of the new determinant remains unchanged , e.g.

$\left| \begin{matrix} {{a}_{11}} & {{a}_{12}} & {{a}_{13}} \\ {{a}_{21}} & {{a}_{22}} & {{a}_{23}} \\ {{a}_{31}} & {{a}_{32}} & {{a}_{33}} \\\end{matrix} \right|=\left| \begin{matrix} {{a}_{11}}+k{{a}_{31}} & {{a}_{12}}+k{{a}_{32}} & {{a}_{13}}+k{{a}_{33}} \\ {{a}_{21}} & {{a}_{22}} & {{a}_{23}} \\ {{a}_{31}} & {{a}_{32}} & {{a}_{33}} \\ \end{matrix} \right|$

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

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

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

11. If $r$ rows (or $r$ columns) become identical, when $\alpha$ is substituted for $x$, then (${\left(x-a\right)}^{r-1}$is a factor of given determinant.

Important Result on Determinants

1. $\left|AB\right|=\left|A\right|\left|B\right|,$ where $A$and$B$ are square matrices of the same order.

2. $\left|A^n\right|={\left|A\right|}^n$

3. If $A,B,$and$C$ are square matrices of the same order such that $i$ th column (or row) of $A$is the sum of $i$ th columns (or rows) of $B$and $C$ and all other columns (or rows)of $A$,$B$and$C$are identical , then$\left|A\right|=\left|B\right|+\left|C\right|$

4. $\left|I_n\right|=1,$ where$I_n$ is identity matrix of order $n.$

5. $\left|O_n\right|=0,$ where $O_n$is a zero matrix of order $n$.

6. If$\Delta \left(x\right)$ be a 3^rd order determinant having polynomials as its elements.

(a) If $\Delta \left(x\right)$ has 2 rows (or columns ) Proportional, then $\left(x-\alpha \right)$ is a factor of $\Delta \left(x\right).$

(b) If$\Delta \left(a\right)$ has 3 rows (or columns ) proportional, then${\left(x-a\right)}^2$ is a factor of$\Delta \left(x\right).$

7. A square matrix $A$ is non-singular, if $\left|A\right|\ne 0$and singular, if $\left|A\right|=0.$

8. Determinant of a skew-symmetric matrix of odd order is zero and of even order is a non-zero perfect square.

9. In general, $\left|B+C\right|\ne \left|B\right|+\left|C\right|$

10. Determinant of a diagonal matrix

= Product of its diagonal elements

11. Determinant of a triangular matrix

= Product of its diagonal elements

12. A square matrix of order $n$ is non-singular, if its rank $r=n$ i.e.if

$\left|A\right|\ne 0,$then rank $\left(A\right)=n$

13. If $\Delta \left(x\right)=\left|\begin{array}{c}{f}_1\left(x\right)f_2\left(x\right)f_3\left(x\right)\\ g_1\left(x\right)g_2\left(x\right)g_3\left(x\right)\\ abc\\ \end{array}\right|,$then

(a) $\sum\limits_{x=1}^{n}{\Delta \left( x \right)=\left| \begin{matrix} \sum\limits_{x=1}^{n}{{{f}_{1}}\left( x \right)} & \sum\limits_{x=1}^{n}{{{f}_{2}}\left( x \right)} & \sum\limits_{x=1}^{n}{{{f}_{3}}\left( x \right)} \\ \sum\limits_{x=1}^{n}{{{g}_{1}}\left( x \right)} & \sum\limits_{x=1}^{n}{{{g}_{2}}\left( x \right)} & \sum\limits_{x=1}^{n}{{{g}_{3}}\left( x \right)} \\ a & b & c \\ \end{matrix} \right|}$

(b) $\prod _{x=1}^n\Delta \left(x\right)=\left|\begin{array}{c}\prod _{x=1}^n{f}_1\left(x\right)\prod _{x=1}^n{f}_2\left(x\right)\prod _{x=1}^n{f}_3\left(x\right)\\ \prod _{x=1}^n{g}_n\left(x\right)\prod _{x=1}^n{g}_2\left(x\right)\prod _{x=1}^n{g}_3\left(x\right)\\ abc\\ \end{array}\right|$

14. If $A$ is a non-singular matrix, then $\left|A^{-1}\right|=\frac{1}{\left|A\right|}={\left|A\right|}^{-1}.$

15. Determinant of a orthogonal matrix $=1or-1.$

16. Determinant of a hermitian matrix and $AB=O,$ then it implies $\left|A\right|=O$and$\left|B\right|=O.$

Minors and Cofactors

If $\Delta =\left|\begin{array}{c}{a}_{11}{a}_{12}{a}_{13}\\ a_{21}{a}_{22}{a}_{23}\\ a_{31}{a}_{32}{a}_{33}\end{array}\right|,$ then the minor $M_{ij}$ of the element $a_{ij}$ the determinant obtained by deleting the $i$th row and jth column.

i.e. $M_{11}=$minor of $a_{11}=\left|\begin{array}{c}{a}_{22}{a}_{23}\\ a_{32}{a}_{33}\end{array}\right|$

$M_{12}=$ minor of $a_{12}=\left|\begin{array}{c}{a}_{21}{a}_{23}\\ a_{31}{a}_{33}\end{array}\right|$

$M_{13}=\left|\begin{array}{c}{a}_{21}{a}_{22}\\ a_{31}{a}_{32}\end{array}\right|$

The cofactor of the element $a_{ij}$ is $C_{ij}={\left(-1\right)}^{i+j}{M}_{ij}$

Properties of Minors and cofactors

1. The sum of the products of elements of any row (or column) of a determinant with the cofactors of the corresponding elements of any other row (or column) is zero, i.e. if

$\Delta =\left|\begin{array}{c}{a}_{11}{a}_{12}{a}_{13}\\ a_{21}{a}_{22}{a}_{23}\\ a_{31}{a}_{32}{a}_{33}\end{array}\right|,$

Then $a_{11}{C}_{31}+a_{12}{C}_{32}+a_{13}{C}_{33}=0$ so on.

2. The sum of the product of elements of any row (or column) of a determinant with the cofactors of the corresponding elements of the same row ( or column) is $\Delta$.

i.e. If$A=\left|\begin{array}{c}{a}_{11}{a}_{12}{a}_{13}\\ a_{21}{a}_{22}{a}_{23}\\ a_{31}{a}_{32}{a}_{33}\end{array}\right|,$ then $\left|A\right|=\Delta =a_{11}{C}_{11}+a_{12}{C}_{12}+a_{13}{C}_{13}$

3. In general, if $\left|A\right|=\Delta ,$ then $\left|A\right|=\sum _{i=1}^n{a}_{ij}{C}_{ij}$

And (add $A$) ${\Delta }^{n-1},$where $A$ is a matrix of order $n\times n.$

Adjoint of a Matrix Adjoint of a matrix is the transpose of the matrix of cofactors of the given matrix, i.e.

$\left(A\right)={\left[\begin{array}{c}{C}_{11}{C}_{12}{C}_{13}\\ C_{21}{C}_{22}{C}_{23}\\ C_{31}{C}_{32}{C}_{33}\end{array}\right]}^T=\left[\begin{array}{c}{C}_{11}{C}_{21}{C}_{31}\\ C_{12}{C}_{22}{C}_{32}\\ C_{13}{C}_{23}{C}_{33}\end{array}\right]$

Properties of Adjoint of a Square Matrix

If $A$and $B$ are square matrices of order $n,$ then

(i) $A$(adj$A$)=(adj$A$) $A=\left|A\right|I$

(ii) adj $\left(A^{\prime }\right)={\left(adjA\right)}^{\prime }$

(iii) adj $\left(AB\right)=\left(adjB\right)\left(adjA\right)$

(iv) adj $\left(kA\right)=k^{n-1}\left(adjA\right),k\in R$

(v) adj $\left(A^m\right)={\left(adjA\right)}^m$

(vi) adj $\left(adjA\right)={\left|A\right|}^{n-2}A,A$ is a non-singular matrix.

(vii) $\left|adjA\right|={\left|A\right|}^{n-1},A$is a non-singular matrix.

(viii) $\left|adjA\left(adjA\right)\right|={\left|A\right|}^{{\left(n-1\right)}^2},$ $A$ is a non-singular matrix.

(ix) Adjoint of a diagonal matrix is a diagonal matrix.

(x) Adjoint of a triangular matrix is a triangular matrix

(xi) Adjoint of a symmetric matrix is a symmetric matrix.

(xii) adj $\left(I_n\right)=I_n,$adj $\left(O\right)=O$

Inverse of Square Matrix

Let $A$ be a square matrix of order $n,$ then a square matrix $B,$ such that $AB=BA=I,$is called inverse of $A$, denoted by $A^{-1}$.

i.e. $A^{-1}=\frac{1}{\left|A\right|}(adjA)$

or $A{A}^{-1}=A^{-1}A=I$

Properties of Inverse of a Square Matirx

(i) Square matrix $A$is invertible, iff it is non-singular.

(ii) ${\left(A^{-1}\right)}^{-1}=A$

(iii) ${\left(A^{\prime }\right)}^{-1}={\left(A^{-1}\right)}^{\prime }$

(iv) ${\left(AB\right)}^{-1}=B^{-1}{A}^{-1}$

In general ,${\left(A_1{A}_2{A}_3............A_n\right)}^{-1}={A_n}^{-1}{A}_{n-1}^{-1}.......A_3^{-1}{A}_2^{-1}{A}_1^{-1}$

(v) If a non-singular square matrix $A$ is symmetric, then $A^{-1}$ is also symmetric.

(vi) $\left|A^{-1}\right|={\left|A\right|}^{-1}$

(vii) $A{A}^{-1}=A^{-1}A=I$

(viii) ${\left(A^k\right)}^{-1}={\left(A^{-1}\right)}^k,k\in N$

(ix) If $A=\left[\begin{array}{c}a00\\ 0b0\\ 00c\\ \end{array}\right]$ and $abc\ne 0,$then $A^{-1}=\left[\begin{array}{c}1/a00\\ 01/b0\\ 001/c\\ \end{array}\right]$.

(x) If $A,B$and $C$ be square matrices of the same order and $A$ is a non-singular matrix, then

(a) $AB=AC\Rightarrow B=C$ [left cancellation law]

(b) $BA=CA\Rightarrow B=C$ [right cancellation law]

(xi) A square matrix is invertible iff it is non-singular and every invertible matrix possesses a unique inverse.

Differentiation of Determinant

If $\Delta \left(x\right)=\left|\begin{array}{c}a\left(x\right)b\left(x\right)c\left(x\right)\\ p\left(x\right)q\left(x\right)r\left(x\right)\\ u\left(x\right)v\left(x\right)w\left(x\right)\end{array}\right|,$

Then $\dfrac{d\Delta }{dx}=\left| \begin{matrix} {a}'\left( x \right) & {b}'\left( x \right) & {c}'\left( x \right) \\ p\left( x \right) & q\left( x \right) & r\left( x \right) \\ u\left( x \right) & v\left( x \right) & w\left( x \right) \\\end{matrix} \right|+\,\,\,\left| \begin{matrix} a\left( x \right) & b\left( x \right) & c\left( x \right) \\ {p}'\left( x \right) & {q}'\left( x \right) & {r}'\left( x \right) \\ u\left( x \right) & v\left( x \right) & w\left( x \right) \\\end{matrix} \right|\,\,+\left| \begin{matrix} a\left( x \right) & b\left( x \right) & c\left( x \right) \\ p\left( x \right) & q\left( x \right) & r\left( x \right) \\ {u}'\left( x \right) & {v}'\left( x \right) & {w}'\left( x \right) \\\end{matrix} \right|$

Integration of Determinant

If $\Delta \left(x\right)\left|\begin{array}{c}{a}_{11}\left(x\right)a_{12}\left(x\right)a_{13}\left(x\right)\\ a_{21}{a}_{22}{a}_{23}\\ a_{31}{a}_{32}{a}_{33}\end{array}\right|,$

Then $\int \Delta \left(x\right)dx=\left|\begin{array}{c}\int a_{11}\left(x\right)dx\int a_{12}\left(x\right)dx\int a_{13}\left(x\right)dx\\ a_{21}{a}_{22}{a}_{23}\\ a_{31}{a}_{32}{a}_{33}\end{array}\right|$

If the elements of more than one column or rows are functions of $x$, then the in integration can be done only after evaluation/expansion of the determinant.