To determine the nature of solutions for a system of linear equations, we typically analyze the determinant of the coefficient matrix and the ranks of the coefficient matrix and the augmented matrix. 

Linear System Analysis

The given system of linear equations is in the matrix form \(AX = B\):

\[\left( {\begin{array}{} 2&1&3\\ 3&0&1\\ 1&2&5 \end{array}} \right)\left( {\begin{array}{} a\\ b\\ c \end{array}} \right) = \left( {\begin{array}{} 5\\ { - 4}\\ {14} \end{array}} \right)\]

Here, the coefficient matrix \(A\), the variable matrix \(X\), and the constant matrix \(B\) are:

\[A = \begin{pmatrix} 2 & 1 & 3 \\ 3 & 0 & 1 \\ 1 & 2 & 5 \end{pmatrix}, \quad X = \begin{pmatrix} a \\ b \\ c \end{pmatrix}, \quad B = \begin{pmatrix} 5 \\ -4 \\ 14 \end{pmatrix}\]

Determinant Calculation

First, we calculate the determinant of the coefficient matrix \(A\), denoted as \(\det(A)\) or \(|A|\).

\[\det(A) = \begin{vmatrix} 2 & 1 & 3 \\ 3 & 0 & 1 \\ 1 & 2 & 5 \end{vmatrix}\]

Using cofactor expansion along the first row:

\[\det(A) = 2 \begin{vmatrix} 0 & 1 \\ 2 & 5 \end{vmatrix} - 1 \begin{vmatrix} 3 & 1 \\ 1 & 5 \end{vmatrix} + 3 \begin{vmatrix} 3 & 0 \\ 1 & 2 \end{vmatrix}\] \[\det(A) = 2((0)(5) - (1)(2)) - 1((3)(5) - (1)(1)) + 3((3)(2) - (0)(1))\] \[\det(A) = 2(0 - 2) - 1(15 - 1) + 3(6 - 0)\] \[\det(A) = 2(-2) - 1(14) + 3(6)\] \[\det(A) = -4 - 14 + 18\] \[\det(A) = -18 + 18\] \[\det(A) = 0\]

Solutions Interpretation based on Determinant

Since the determinant of the coefficient matrix \(A\) is \(0\), this tells us that the matrix \(A\) is singular. A singular coefficient matrix means that the system of linear equations does not have a unique solution. Instead, it will either have:

• No solution (inconsistent system)
• Infinitely many solutions (consistent system with dependent equations)

To distinguish between these two cases, we need to analyze the ranks of the coefficient matrix \(A\) and the augmented matrix \([A|B]\).

Augmented Matrix and Rank Analysis

We form the augmented matrix \([A|B]\) by combining the coefficient matrix \(A\) with the constant vector \(B\):

\[[A|B] = \begin{pmatrix} 2 & 1 & 3 & | & 5 \\ 3 & 0 & 1 & | & -4 \\ 1 & 2 & 5 & | & 14 \end{pmatrix}\]

Now, we perform elementary row operations to transform the augmented matrix into its row echelon form to find its rank.

From the row echelon form, we can determine the ranks:

• The rank of the coefficient matrix \(A\), denoted as \(\rho(A)\), is the number of non-zero rows in the coefficient part of the row echelon form. Here, \(\rho(A) = 2\).
• The rank of the augmented matrix \([A|B]\), denoted as \(\rho([A|B])\), is the number of non-zero rows in the entire row echelon form. Here, \(\rho([A|B]) = 2\).

Consistency and Solutions Conclusion

According to the Rouché–Capelli theorem (also known as the Frobenius theorem), the consistency and nature of solutions for a system of linear equations \(AX=B\) are determined by comparing the ranks of \(A\) and \([A|B]\) with the number of variables \(n\):

• If \(\rho(A) \neq \rho([A|B])\), the system is inconsistent and has no solution.
• If \(\rho(A) = \rho([A|B])\), the system is consistent.
  • If \(\rho(A) = \rho([A|B]) = n\) (number of variables), the system has a unique solution.
  • If \(\rho(A) = \rho([A|B]) < n\), the system has infinitely many solutions.

In our case:

• \(\rho(A) = 2\)
• \(\rho([A|B]) = 2\)
• Number of variables \(n = 3\) (a, b, c)

Since \(\rho(A) = \rho([A|B]) = 2\) and this rank is less than the number of variables \(n = 3\), the system of linear equations has infinitely many solutions.