Matrices & Determinants
Similar Matrices
Grade 12
Question:
<p>\(P\) is a non-singular matrix and \(A\), \(B\) are two matrices such that \(B = P^{-1}AP\). The true statements among the following are</p>
<p>\(A\) is invertible iff \(B\) is invertible</p>
<p>\(B^n = P^{-1}A^n P\) \(\forall\, n \in N\)</p>
<p>\(\forall\, \lambda \in R\), \(B\), \(\lambda I - P^{-1}(\lambda I)P\)</p>
<p>\(A\) and \(B\) are both singular matrices</p>
Step-by-Step Solution
Key Concept: Similar matrices B = P⁻¹AP share fundamental properties like eigenvalues, trace, determinant, and rank, because similarity transformations preserve these invariants under change of basis.
<p><strong>Key Property:</strong> If B = P⁻¹AP, then A and B are similar matrices.</p><p><strong>Properties Preserved Under Similarity:</strong></p><p><strong>✓ Same Eigenvalues:</strong> det(B - λI) = det(P⁻¹AP - λI) = det(P⁻¹(A - λI)P) = det(P⁻¹)·det(A - λI)·det(P) = det(A - λI)</p><p><strong>✓ Same Trace:</strong> tr(B) = tr(P⁻¹AP) = tr(AP·P⁻¹) = tr(A)</p><p><strong>✓ Same Determinant:</strong> det(B) = det(P⁻¹AP) = det(P⁻¹)·det(A)·det(P) = det(A)</p><p><strong>✓ Same Rank:</strong> rank(B) = rank(P⁻¹AP) = rank(A) (left/right multiplication by non-singular matrices preserves rank)</p><p><strong>✓ Same Characteristic Polynomial:</strong> Both have identical characteristic equations</p><p><strong>Properties NOT Preserved:</strong></p><p><strong>✗ Eigenvectors are different:</strong> If v is an eigenvector of A, then P⁻¹v is the corresponding eigenvector of B</p><p><strong>✗ Matrix entries differ:</strong> A and B have different element values unless P = cI</p><p>∴ Answer: Statements about eigenvalues, trace, determinant, rank, and characteristic polynomial are TRUE. Statements claiming identical entries or eigenvectors are FALSE.</p>
Correct Answer: A,B