Exercises 21–26, find (a) a basis for the column space and (b) the rank of the matrix. 2 4 -3 7 14 6 -3 25. -4 1 -2 4 -2 -2] 2.
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- CAPSTONE Explain how to determine whether an nn matrix A is diagonalizable using a similar matrices, b eigenvectors, and c distinct eigenvalues.Proof Prove that if A and B are similar matrices and A is nonsingular, then B is also nonsingular and A1 and B1 are similar matrices.True or False? In Exercises 37 and 38, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. a If A and B are similar nn matrix, then they have always the same characteristics polynomial equation. b The fact that an nn matrix A has n distinct eigenvalues does not guarantee that A is diagonalizable.
- Proof Let A be an mn matrix. a Prove that the system of linear equations Ax=b is consistent for all column vectors b if and only if the rank of A is m. b Prove that the homogeneous system of linear equations Ax=0 has only the trivial solution if and only if the columns of A are linearly independent.Proof Let A be an nn square matrix. Prove that the row vectors of A are linearly dependent if and only if the column vectors of A are linearly dependent.Proof Prove that row operations do not change the dependency relationships among the columns of an mn matrix.