Question

If $\rho \left( A \right)=\rho \left( A,B \right)$ then the system is\[\]
A. Consistent and has infinitely many solution\[\]
B. Consistent and has unique solution\[\]
C. Consistent \[\]
D. Inconsistent \[\]

Answer 1

Hint:
 We recall the definitions of rank of a matrix $\rho $ , coefficient matrix $A$ and augmented matrix $\left( A,B \right)$. We use the fact that the system is consistent and has a unique solution if the constant vector in the augmented matrix can be expressed as a linear combination of coefficients in other words $\rho \left( A \right)=\rho \left( A,B \right)$.\[\]

 Complete step by step answer:
We know that the standard form of system of linear equations with an $n$ variables is given as
\[\begin{align}
  & {{a}_{11}}{{x}_{1}}+{{a}_{12}}{{x}_{2}}+...+{{a}_{1n}}{{x}_{n}}={{b}_{1}} \\
 & {{a}_{21}}{{x}_{1}}+{{a}_{22}}{{x}_{2}}+...+{{a}_{2n}}{{x}_{n}}={{b}_{2}} \\
 & \vdots \\
 & {{a}_{n1}}{{x}_{1}}+{{a}_{n2}}{{x}_{2}}+...+{{a}_{nn}}{{x}_{n}}={{b}_{n}} \\
\end{align}\]
We can write the above system of equations in the matrix from as
\[\begin{align}
  & \left[ \begin{matrix}
   {{a}_{11}} & {{a}_{12}} & \cdots & {{a}_{1n}} \\
   {{a}_{21}} & {{a}_{22}} & \cdots & {{a}_{2n}} \\
   \vdots & \vdots & {} & {} \\
   {{a}_{n1}} & {{a}_{n2}} & \cdots & {{a}_{nn}} \\
\end{matrix} \right]\left[ \begin{matrix}
   {{x}_{1}} \\
   {{x}_{2}} \\
   \vdots \\
   {{x}_{n}} \\
\end{matrix} \right]=\left[ \begin{matrix}
   {{b}_{1}} \\
   {{b}_{2}} \\
   \vdots \\
   {{b}_{n}} \\
\end{matrix} \right] \\
 & \Rightarrow AX=B \\
\end{align}\]
We have denoted the square matrix containing the coefficients of the variables called coefficient matrix as $A$, the column vector containing the variables as $X$ and the column vector containing the constant terms as called constant vector as$B$. The augmented matrix which augments the matrix $A$ and $B$ is denoted as $\left( A,B \right)$ and written as
\[\left[ \begin{matrix}
   {{a}_{11}} & {{a}_{12}} & {} & {{a}_{1n}} & {{b}_{1}} \\
   {{a}_{21}} & {{a}_{22}} & {} & {{a}_{2n}} & {{b}_{2}} \\
   {} & {} & {} & {} & {} \\
   {{a}_{n1}} & {{a}_{n2}} & {} & {{a}_{nn}} & {{b}_{n}} \\
\end{matrix} \right]\]
The rank of matrix is denoted $\rho $ is the number of linearly independent column vectors or we can say the number vectors that cannot be expressed as a linear of combination of other vectors, for example we observe the following matrix that is
\[A=\left[ \begin{matrix}
   1 & 2 & 3 \\
   3 & 4 & 7 \\
   6 & 5 & 9 \\
\end{matrix} \right]\]
We observe that none of the column vectors can be expressed as linear combination of other two . So number of linearly independent vectors is 3 and $\rho \left( A \right)=3$. If we replace the third column with ${{C}_{3}}={{\left[ \begin{matrix}
   3 & 7 & 11 \\
\end{matrix} \right]}^{T}}$ and then we have ${{C}_{3}}={{C}_{1}}+{{C}_{2}}$ expressed as linear combination. So the rank will reduce to 2. If the rank is same dimension of matrix we call it full-rank matrix. \[\]

We know that a system is consistent when $\det \left( A \right)\ne 0$ which can only happen when the matrix is of full rank $\rho \left( A \right)=n$. The system $AX=B$ will be consistent and have unique solution when the column vector of constants in the augmented matrix can expressed as a linear combination of coefficients. So we will have $\rho \left( A \right)=\rho \left( A,B \right)=n$. So the correct option is B. We find a unique solution from the following system of equations because the ranks of coefficient and augmented matrix are equal.
\[\left[ \begin{matrix}
   1 & 2 & 3 \\
   3 & 4 & 7 \\
   6 & 5 & 9 \\
\end{matrix} \right]\left[ \begin{matrix}
   x \\
   y \\
   z \\
\end{matrix} \right]=\left[ \begin{matrix}
   0 \\
   2 \\
   11 \\
\end{matrix} \right]\]

Note:
The system will be consistent and will have infinitely many solution if $\rho \left( A \right)<\rho \left( A,B \right)$. We note that the system will be inconsistent if there is no solution and will become indeterminate if there is more than one solution.