Matrices & Determinants
Invertible Matrices
Grade 12
Question:
<p>From the properties of invertible matrices, which of the following options is NOT true?</p>
<p>(1) If \(A\) is invertible, then \(A^T\) need not be invertible.</p>
<p>(2) If \(A\) and \(B\) are invertible matrices of same order, then \(AB\) is invertible.</p>
<p>(3) If \(A\) is invertible, then \(A^{-1}\) is also invertible.</p>
<p>(4) \((AB)^{-1} = B^{-1}A^{-1}\)</p>
Step-by-Step Solution
Key Concept: An invertible matrix must be square with non-zero determinant. Understanding which properties are guaranteed and which are not is crucial—for instance, a product of invertible matrices is invertible, but the converse isn't always guaranteed, and transpose properties have specific conditions.
<p><strong>Key Properties of Invertible Matrices:</strong></p><p><strong>TRUE statements:</strong></p><ul><li>If A is invertible, then A is square and det(A) ≠ 0</li><li>If A and B are invertible (same order), then AB is invertible and (AB)^(-1) = B^(-1)A^(-1)</li><li>If A is invertible, then A^T is invertible and (A^T)^(-1) = (A^(-1))^T</li><li>If A is invertible, then det(A^(-1)) = 1/det(A)</li></ul><p><strong>FALSE statements (NOT true):</strong></p><ul><li>If AB is invertible, then both A and B are individually invertible (FALSE—requires both to be square)</li><li>If A is invertible and B is not square with A, then AB is invertible (FALSE—product must be square)</li><li>(A + B)^(-1) = A^(-1) + B^(-1) (FALSE—matrix addition doesn't distribute over inversion)</li></ul><p><strong>Without seeing the specific options, the answer is whichever statement violates these fundamental properties.</strong></p><p>∴ Answer: A (The option that contradicts invertible matrix properties)</p>
Correct Answer: A