If the rank of an n x n matrix A is (n - 1), then the system of equations Ax= b hasa)(n - 1) parameter family of solutionsb)one parameter family of solutionsc)no solutiond)a unique solutionCorrect answer is option 'B'. Can you explain this answer?

Question

Answer

Explanation:

To understand why the correct answer is option B, let's break down the problem step by step.

Rank of a matrix:
The rank of a matrix is defined as the maximum number of linearly independent rows or columns in the matrix. In other words, it is the dimension of the vector space spanned by the rows or columns of the matrix.

System of equations:
The system of equations Ax = b represents a set of linear equations, where A is the coefficient matrix, x is the unknown variable vector, and b is the constant vector.

n x n matrix with rank (n - 1):
Given that the rank of the n x n matrix A is (n - 1), it means that there are (n - 1) linearly independent rows or columns in matrix A. Since A is a square matrix, it implies that one row or column is a linear combination of the remaining (n - 1) rows or columns.

Solving the system of equations:
Now, let's consider the system of equations Ax = b, where A is an n x n matrix with rank (n - 1). Since one row or column of A is a linear combination of the remaining (n - 1) rows or columns, it means that there is redundancy or dependency in the system of equations.

This redundancy or dependency results in infinitely many solutions or a parameter family of solutions. In other words, there is not a unique solution to the system of equations.

Conclusion:
Therefore, the correct answer is option B) one parameter family of solutions. This means that the system of equations Ax = b has infinitely many solutions, which can be expressed using a single parameter. Each value of the parameter corresponds to a different solution in the parameter family.

Answer

Rank(A)+rank(solution space) =n

n-1+rank(solution space)=n

rank(solution space) =1

hence, it has one parameter family of solution

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