Let A=( a 0 b 2 a 2 a b b 2 a a ) be the aug

step 1 We have given the matrix here 1, 2 and minus 4, 0A equals to B. So now we have to find here, det

Let A=( a 0 b 2 a 2 a b ...

Let $A=\left(\begin{array}{ccc|c}a & 0 & b & 2 \\ a & 2 & a & b \\ b & 2 & a & a\end{array}\right)$ be the augmented matrix for a linear system. For which values of $a$ and $b$ does the system have (i) a unique solution? (ii) infinitely many solutions? (iii) no solution?

Key Concepts

Classification of Solutions in Linear Systems

Linear systems can be classified based on their solution characteristics: a unique solution arises when the coefficient matrix is invertible (nonzero determinant), infinitely many solutions occur when the system is consistent but underdetermined (often seen when the determinant is zero and there are free variables), and no solution results when the system is inconsistent – typically when there is a contradiction in the equations.

Consistency and Inconsistency

A system of linear equations is considered consistent if there is at least one solution, and inconsistent if there is no solution. When the rank of the coefficient matrix equals the rank of the augmented matrix, the system is consistent; if these ranks differ, the system is inconsistent, leading to no solution.

Rank of a Matrix

The rank of a matrix is the maximum number of linearly independent rows (or columns) in the matrix. In the context of linear systems, comparing the rank of the coefficient matrix with that of the augmented matrix helps determine whether the system is consistent and whether it has a unique solution or infinitely many solutions.

Coefficient Matrix and Determinant

The coefficient matrix is the portion of an augmented matrix that contains only the variables’ coefficients. Its determinant helps determine if the matrix is invertible; a nonzero determinant indicates a unique solution for the system, while a zero determinant suggests the possibility of either infinitely many solutions or no solution.

Augmented Matrix

An augmented matrix compiles the coefficients and constant terms of a system of linear equations into one matrix. This compact form is used for systematic operations like Gaussian elimination to determine the nature of the solutions of the system.

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