PYQs based on Rank of Matrix | Matrix скачать в хорошем качестве

PYQs based on Rank of Matrix | Matrix

This video from Grad Math Mentor (0:00) provides a detailed overview of various questions asked in competitive exams related to the rank of a matrix. The presenter solves each problem step-by-step, explaining the concepts and formulas used to determine the rank of a matrix. Here's a breakdown of the key topics and examples covered: • Calculating the rank of a matrix using elementary row operations (0:44). The video demonstrates how to transform a matrix into an equivalent form to find the number of non-zero rows. • Rank of a non-singular matrix (1:13). It's explained that for a non-singular matrix of order n, its rank is also n, based on the minor method. • Rank of an identity matrix (2:07). The rank of an identity matrix of order n is shown to be n. • Impact of identical rows on matrix rank (2:34). If a matrix has two identical rows, its rank will be reduced, resulting in at least one non-zero row. • Effect of elementary row operations on rank (3:50). The video clarifies that elementary row operations do not change the rank of a matrix. • Rank of a zero matrix (4:18). A zero matrix always has a rank of zero. • Maximum rank of a matrix based on its order (4:50). The maximum possible rank of a matrix is determined by the minimum of its number of rows and columns. • Relationship between rank and linearly independent rows (5:22). The rank of a matrix is equal to the number of its linearly independent rows. • Properties of ranks in matrix multiplication and addition (7:08). The video discusses formulas for the rank of a product of matrices (rank(AB) ≤ min(rank(A), rank(B))) and the rank of a sum of matrices (rank(A+B) ≤ rank(A) + rank(B)). • Rank of a non-singular matrix multiplied by another matrix (7:30). If A is non-singular, rank(AB) = rank(B) and rank(A⁻¹B) = rank(B).