Consider the following statements about the matrix M = pmatrix 71 & 23 & 48 \\ 57 & 28 & 29 \\ 65 & 17 & 48 pmatrix : Statement-I: The inverse of M does not exist. Statement-II: M is non-singular. Which one of the following is correct in respect of the above statements?
C. Statement-I is correct but Statement-II is not correct
The given matrix is M = pmatrix 71 & 23 & 48 \\ 57 & 28 & 29 \\ 65 & 17 & 48 pmatrix . Let C_1, C_2, C_3 be the first, second, and third columns of the matrix M respectively. Observing the elements of the columns, we get: C_2 + C_3 = pmatrix 23 + 48 \\ 28 + 29 \\ 17 + 48 pmatrix = pmatrix 71 \\ 57 \\ 65 pmatrix = C_1 Since the first column is the sum of the second and third columns, the columns of M are linearly dependent. Applying the column operation C_1 C_1 - (C_2 + C_3), the first column becomes a zero column. Therefore, the determinant of the matrix M is |M| = 0. Since the determinant is zero, M is a singular matrix. Thus, the inverse of M does not exist. Statement-I is correct because the inverse of M does not exist. Statement-II is not correct because M is a singular matrix. Answer: Statement-I is correct but Statement-II is not correct
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