A is a 3 x 4 real matrix and A x = b is an inconsistent system of equations. The highest possible rank of A isa)1b)2c)3d)4Correct answer is option 'B'. Can you explain this answer?

Question

Answer

Highest possible rank of A= 2 ,as Ax = b is an inconsistent system.

Answer

Since a is 3*4 matrix hence x has 4 variables therefore highest rank is 3

Answer

2

Answer

Explanation:
To find the highest possible rank of a matrix, we need to consider the row-echelon form of the matrix. The row-echelon form is a simplified version of the matrix where the leading entry (the first non-zero entry) of each row is to the right of the leading entry of the row above it.

Rank of a matrix:
The rank of a matrix is the number of non-zero rows in its row-echelon form. It represents the maximum number of linearly independent rows or columns in the matrix.

Inconsistent system of equations:
An inconsistent system of equations is a system of equations that does not have a solution. In matrix form, it can be represented as A x = b, where A is the coefficient matrix, x is the variable vector, and b is the constant vector. If the system is inconsistent, it means that there are contradictory equations or the equations are not solvable.

Rank of the coefficient matrix:
The rank of the coefficient matrix A determines the number of independent equations in the system. If the rank of A is less than the number of variables (columns), it means that there are redundant or dependent equations, and the system is consistent. If the rank of A is equal to the number of variables, it means that there are no redundant equations, and the system is consistent. If the rank of A is greater than the number of variables, it means that there are contradictory equations, and the system is inconsistent.

A is a 3 x 4 real matrix:
Since A is a 3 x 4 matrix, it means that it has 3 rows and 4 columns.

Highest possible rank of A:
To determine the highest possible rank of A, we need to consider the row-echelon form of A. In the row-echelon form, the leading entry of each row is to the right of the leading entry of the row above it. Since A has 3 rows, the highest possible rank of A can be at most 3.

Inconsistent system of equations:
The system of equations A x = b is inconsistent, which means that there are contradictory equations or the equations are not solvable. In the context of matrices, this means that the rank of A is greater than the rank of the augmented matrix [A | b]. Since the highest possible rank of A is 3, it means that the rank of [A | b] must be greater than 3 for the system to be inconsistent.

Conclusion:
The highest possible rank of A is 3, which means that option 'C' is the correct answer.

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