Final answer:
True statements for all invertible matrices a and b are that 4a is invertible, the conjugate power identity (aba⁻¹)⁹=ab⁹a⁻¹ holds, and a is invertible. False statements include the power of a product (aa⁻¹)⁵=a⁵a⁻⁵, the false distributive property (ab)²=a²b²+2ab, and the non-commutative property ab=ba.
Step-by-step explanation:
When evaluating statements about invertible n×n matrices, a and b, we must apply the rules of matrix algebra. Among the given statements, here are the true ones:
- B. 4a is invertible - Multiplying an invertible matrix by a non-zero scalar yields another invertible matrix.
- D. (aba⁻¹)⁹=ab⁹a⁻¹ - Here, we apply the property of conjugation. When we conjugate matrix b by matrix a (i.e., perform the operation aba⁻¹), raising this product to any power results in ab raised to that power, followed by a raised to the negative of that power.
- E. a is invertible - As stated by the problem, matrix a is invertible by definition.
The other statements are not necessarily true for all invertible matrices:
- A. (aa⁻¹)⁵=a⁵a⁻⁵ - This is false because (aa⁻¹)⁵ = I, where I is the identity matrix, and I raised to any power is I.
- C. (ab)²=a²b²+2ab - This suggests a distributive property which does not hold for matrix multiplication; (ab)²=(ab)(ab)=a(bb)a, assuming (bb) is the identity matrix.
- F. ab=ba - In general, matrix multiplication is not commutative; ab does not necessarily equal ba.