Rank of Matrix Video Lecture | Mathematics for Competitive Exams

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FAQs on Rank of Matrix Video Lecture - Mathematics for Competitive Exams

1. What is the rank of a matrix?
Ans. The rank of a matrix is the maximum number of linearly independent rows or columns in the matrix. It represents the dimension of the vector space spanned by the rows or columns of the matrix.
2. How can the rank of a matrix be calculated?
Ans. The rank of a matrix can be calculated by performing row operations to reduce the matrix to its row-echelon form or reduced row-echelon form. The number of non-zero rows in the row-echelon form or reduced row-echelon form will give the rank of the matrix.
3. Can the rank of a matrix be greater than its number of rows or columns?
Ans. No, the rank of a matrix cannot be greater than its number of rows or columns. The rank of a matrix is always less than or equal to the minimum of the number of rows and columns in the matrix.
4. What does a matrix with full rank mean?
Ans. A matrix is said to have full rank if its rank is equal to the minimum of its number of rows and columns. In other words, a matrix with full rank has linearly independent rows and columns, and its columns span the entire column space.
5. How does the rank of a matrix affect its solutions in a system of linear equations?
Ans. The rank of a matrix is closely related to the number of solutions in a system of linear equations. If the rank of the coefficient matrix is equal to the rank of the augmented matrix, the system has a unique solution. If the rank of the coefficient matrix is less than the rank of the augmented matrix, the system has infinitely many solutions. If the rank of the coefficient matrix is less than the number of variables, the system has no solution.
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