Find the rank of the $\left[\begin{array}{ccc}1& 2& -1\\ 3& -1& 2\\ 1& -2& 3\\ 1& -1& 1\end{array}\right]$ by row reduction method.

1. Adjoint of a Square Matrix:

(i) Definition in Statement Form:

If $A=\left[a_{\mathrm{ij}}\right]$ is a square matrix of order $n$ and $C_{\mathrm{ij}}$ denote the cofactor of $a_{\mathrm{ij}}$ in $A$, then the transpose of the matrix of cofactors of elements of $A$ is called the adjoint of $A$ and is denoted by $\mathrm{adj}A$ i.e. $\mathrm{adj}A\mathit{=}{\left[{\mathit{C}}_{\mathit{ij}}\right]}^{\mathit{T}}$.

(ii) Definition in Mathematical form:

If $A=\left[\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right]$, then, $\mathrm{adj}A=\left[\begin{array}{ccc}{C}_{11}& C_{12}& C_{13}\\ C_{21}& C_{22}& C_{23}\\ C_{31}& C_{32}& C_{33}\end{array}\right]$ where $C_{\mathrm{ij}}$ denotes the cofactor of $a_{\mathrm{ij}}$ in $A$.

2. Adjoint of a Square Matrix of order $\mathbf{2}\mathbf{:}$

(i) It can be obtained by interchanging the diagonal elements and changing the signs of off-diagonal elements.

(ii) If $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$, then $\mathrm{adj}A=\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]$.

3. Following are some properties of the adjoint of a square matrix:

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

(i) $\mathit{adj}\left(AB\right)=\left(adjB\right)\left(adjA\right)$

(ii) $adj{A}^T=(adjA{)}^T$

(iii) $adj\left(adjA\right)=|A{|}^{n-2}A$

(iv) $\left|adjA\right|=|A{|}^{n-1}$

(v) $\left|adj\right(adjA\left)\right|=|A{|}^{(n-1{)}^2}$

(vi) If $A$ is a square matrix of order $n$, then $A\left(adjA\right)=\left|A\right|I_n=\left(adjA\right)A$

4. Inverse of a Matrix:

A square matrix $A$ of order $n$ is invertible if there exists a square matrix $B$ of the same order such that $AB=I_n=BA$. In such a case, we say that the inverse of $A$ is $B$ and we write $A^{-1}=B$.

5. Properties of Inverse of a Matrix:

(i) Every invertible matrix possesses a unique inverse.

(ii) If $A$ is an invertible matrix, then ${\left(A^{-1}\right)}^{-1}=A$.

(iii) A square matrix is invertible if it is non-singular.

(iv) If $A$ is a non-singular matrix, then $A^{-1}=\frac{1}{\left|A\right|}\left(adjA\right)$

(v) If $A$ and $B$ are two invertible matrices of the same order, then $(AB{)}^{-1}=B^{-1}{A}^{-1}$

(vi) The inverse of an invertible symmetric matrix is a symmetric matrix.

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

6. Elementary Matrix:

A matrix obtained from an identity matrix by a single elementary operation is called an elementary matrix.

7. Elementary Transformations:

(i) Interchange of any two rows (columns).

(ii) Multiplying all elements of a row (column) of a matrix by a non-zero scalar.

(iii) Adding to the elements of a row (column), the corresponding elements of any other row (column) multiplied by any scalar$k$.

8. Steps to find the Inverse of a Matrix by Elementary Transformations:

(i) Obtain the square matrix, say$A$. 

(ii) Write $A=I_{n}A$

(iii) Perform a sequence of elementary row operations successively on $A$ on the LHS and the pre-factor $I_n$ on the RHS till we obtain the result $I_n=BA$.  

(iv) Write $A^{-1}=B$.

9. Orthogonal Matrix:

When the product of a square matrix and its transpose gives an identity matrix, then the square matrix is known as an orthogonal matrix i.e., $A{A}^T=I$.

10. Row-Echelon form:

A matrix is in row echelon form if

(i) All rows consisting of only zeroes are at the bottom.

(ii) The leading coefficient (also called the pivot) of a nonzero row is always strictly to the right of the leading coefficient of the row above it.

11. Rank of a Matrix:

A positive integer $r$ is said to be the rank of a non-zero matrix $A$, if

(i) there exists at least one matrix in $A$ of order which is not zero.

(ii) Every Minor in $A$ of order greater than $r$ is zero.

It is written as $\rho \left(A\right)=r$.

12. Properties of Rank of a Matrix:

(i) The rank of every non-singular matrix of order $n$ is $n$.

(ii) The rank of a square matrix $A$ of order $n$ can be less than $n$ iff $\left|A\right|=0$.

(iii) Elementary transformation do not alter the rank of a matrix.

(iv) The rank of a matrix $A$ does not change by pre-multiplication or post-multiplication with any non-singular matrix.

(v) If $A$ and $B$ are matrices of same order, then $\rho \left(AB\right)\le \min \left\{\rho \left(A\right),\rho \left(B\right)\right\}$.

(vi) A square matrix $A$ of order $n$ has inverse if and only if $\rho \left(A\right)=n$.

(vii) The rank of a non-zero matrix is equal to the number of non-zero rows in a row-echelon form of the matrix.

13. Types of Simultaneous Equations:

(i) Consistent Equation:

A system of equations is said to be consistent if they have one or more solutions i.e., unique solution and infinite solutions.

(ii) Inconsistent Equation:

If a system of equations has no solution, it is said to be inconsistent.

14. Solution to a System of Linear Equations:

(i) Matrix Inversion Method:

(a) Consider the matrix equation, $AX=B$.

(b) Pre-multiply both sides of the equation by $A^{-1}$   

(c) $X=A^{-1}B$ is the required solution of the given equations.

(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.

(iii) Gaussian Elimination Method:

In this method, we transform the augmented matrix of the system of linear equations into row-echelon form and then by back-substitution, we get the solution.

15. Consistency of System of Linear Equations by Rank Method:

(i) Non-Homogeneous Equation $AX=B$:

(a) The given system of equation is consistent and has unique solution if $\rho \left(A:B\right)=\rho \left(A\right)=$number of variables

(b) The given system of equation is consistent and has infinitely many solutions if $\rho \left(A:B\right)=\rho \left(A\right)<$number of variables.

(c) The given system of equation is inconsistent if $\rho \left(A:B\right)\ne \rho \left(A\right)$.

(ii) Homogeneous Equation (always consistent) $AX=0$:

(a) The given system of equation is consistent and has trivial (unique) solution if $\rho \left(A\right)=$number of variables.

(b) The given system of equation is consistent and has non-trivial (infinitely many) solution if $\rho \left(A\right)<$number of variables.